Your predicament is a pretty common one, and applies to any GM who has a player focused on summoning monsters. Even without the brokenness that is Master Summoner, summoning-focused characters can quickly break the action economy and make it so the rest of the group has little to do, most of the time. That said, here are a few things that you can do to reduce the power of your Master Summoner player without removing his flavour. There are basically two things to do here: carefully limit the creatures that Summon Monster is capable of summoning, and make the Master Summoner's resources matter more.
Don't add monsters to the Summon Monster list without careful consideration
The list of monsters that you can summon with Summon Monster is reasonably well-balanced as it is. However, there are a bunch of unstated assumptions that it has that can cause some serious problems if they aren't adhered to. Some of your problems come from this.
Ban Summon Good Monster
That feat is really super powerful, especially at lower levels. It gives you way more battlefield control options than you would normally get, as you have already noted. Most of the monsters on that list are fine, but no one is going to summon the well-balanced monsters off a list like that, they're going to summon the awesome, way-too-powerful ones. If you don't want to ban this list outright, then at least go through it with a fine-tooth comb and assure that the abilities that you get from it at a particular level aren't out of line with what you get from the normal list. For example, the first creatures that have any abilities that are either ranged or spells are in the Summon Monster 3 list, and those creatures are pretty terrible is a stand up fight (Dretch and Lantern Archon).
Ban non-standard Elementals
This is similar to the point above. When Summon Monster was first written, there were 4 elementals: Fire, Water, Earth, Air. Each of them is an effective tank in their environment, and are reasonably well-balanced. Allowing the player to summon the other 8 types of elementals that have been added since is adding to his versatility by a significant amount, and versatility equals power. As you've noted, some of the additional elementals have abilities out of line with what the normal elementals have. You don't have to allow the player to summon more powerful monsters (like the Mud Elemental) if it's going to mess up your game.
Keep the general Summon Monster guidelines in mind when adding new monsters
To sort of go back on what I said above, there's nothing wrong with letting your player have more or different creatures to summon, as long as you're careful about what these new creatures give the player at that level. Make sure you look carefully at what a particular level of Summon Monster gives you before adding new things to that level. For example, with Summon Monster 2, you have the following roles: melee ground tank (most of them), flying tank (air elemental), swimming tank (water elemental, octopus, squid), high damage/poison tank (fire elemental, giant centipede, giant spider). The only battlefield control options available to any of these creatures are the normal combat maneuvers, mainly grapple and trip. None of them have more powerful control options like the faun or mud elemental do. This extends through most of the levels of Summon Monster. For example, nothing below Summon Monster 9 has the ability to cast spells like a PC.
The final arbiter of what the player is allowed to summon or not is you, as the GM. If you think that a particular form is out of line for the power that a level of Summon Monster gives, then you are well within your rights to ban that form, or put it on a different level of Summon Monster. It doesn't matter if the rules say that the player can summon a particular monster; if using a particular summon is going to make the game less fun, then that summon should be banned, or otherwise limited.
Make the Master Summoner's resources matter.
The standard game rules make some assumptions about the kind of game that you're playing, and tune player resources based on these assumptions. The game assumes that the average player is going to have 4 encounters in an average day. For a Master Summoner with 18 Charisma, that means that they can summon 2 monsters in each encounter, and 3 in one of them. Adding 2 monsters to a fight is something that can be worked around, in general. If you have significantly fewer encounters, then you need to figure out a way to make your player's resources matter again.
Intelligent enemies should have intelligent protections.
For example: Any intelligent enemy that knows about the summoner is going to have a Protection from Good ready to go (whether from a potion, and item, a scroll, or a spell prepared, depending on the enemy). Protection from Good doesn't shut down your player entirely, but it does make it so he needs to use more of his Summon Monster abilities in a single combat to remain effective. Spellcasters can use Dispel Magic to quickly end a summon. More prepared enemies can use Magic Circle instead of Protection, which is a little more effective.
Reduce the number of summoned monsters at once.
Like you say at the end of your question, you might consider limiting the number of monsters summoned to two spells. The player could use one eidolon and one Summon Monster spell, or two Summon Monster spells, but no more. This means that the player can basically use the same number of Summon Monster spells that he could normally use per encounter in a 4-encounter-per-day game. You might allow the player to use extra uses of the ability to get more monsters at a time, to make the player feel like those extra uses are still useful. For example, maybe the first 2 castings take one use each, but any after that take 3 each.
Reduce the number of uses per day.
If your campaign requires only one or two encounters per day, then it might be helpful to reduce the number of Summon Monster uses that your player gets directly. In a 2 encounter per day game, reducing the number of uses to 5 or 6 would likely be helpful. This might make your player feel shafted, so it might not be the best thing to do.
Use more numerous monsters, or ranged monsters.
Basically, alter your encounter design to take the summoner into account. If there are a dozen orcs charging in that will likely overwhelm the party, the summoner can deal with them while the party deals with the rest of the encounter. Alternately, powerful ranged monsters can target your summoner, making it a priority for the summoner to send minions to deal with that threat while the other players take out the main encounter.
Melee-centric battle priest? I think they're called "paladins..."
If you don't have a paladin in the party (which I suspect when you say no one is in heavy armor), then you're going to miss out on a lot of goodies. We have 3 paladins, an oracle, a priest, and a sorcerer in our WotR party and the treasure, benefits, etc. for paladin types more than make up for whatever build steps you're considering. Go paladin of Iomedae and you'll be geared with legendary items beyond your wildest dreams. Torag is a good second choice as there's a fair number of Torag shrines and stuff in chapters 1 and 2 at least (which is where we are). The GM is keeping us one character level behind where they say you're supposed to be and it's still been mostly on easy mode with that class mix.
Two-hander is always strictly better than one-hander and shield unless you've cobbled together just the right set of feats and stuff from fringe splatbooks to set something particular up (two-handed fighting with bull rushes from the shield slams) but if you want to bring the damage you should stay two-hander.
Best Answer
Acquiring Plant Allies Is Difficult without D&D 3.5 Materials
Getting plant allies as a cleric--even if the cleric has the plant Domain--is just not a well supported option in Pathfinder. Summoning them appears impossible, and even making Diplomacy skill checks versus low-Intelligence plants (wild plants?) requires the wild empathy ability then the feat Greater Wild Empathy, and unlike so many things in Pathfinder (like the once class-feature-only ability to use the Disable Device skill to disable magical traps that can now be gotten with a trait) wild empathy is--so far--only ever offered as a class feature, requiring multiclassing into druid or ranger or whatever.
Options
There are only a handful of options, and none of them are what you want, but maybe you can make do.
The spell awaken has been revised and clarified from its D&D 3.5 roots to make the affected creature more like a follower than just a new sentient who digs you. But the spell's not on your spell list.
The Staff of Because Talking Plants Are Awesome
Aura: Transmutation and abjuration; CL: 9th; Slot: None; Price: 12,070 gp; Weight: 5 lbs.
Description: This hollow steel staff depicts clerics in metal armor carrying light steel shields and wielding morningstars beating up sobbing hippies. When used as a weapon, it makes a deeply satisfying hollow clang that seems louder when the victim is a druid. The staff allows use of the following spells:
Construction Requirements: Craft Staff, awaken, shield of faith. Cost: 6,035 gp (Math: ((400 gp x 5 for the spell awaken x 9 for the caster level) + (2,000 gp for material components x 50 to charge the staff) / 10 for 10 charges per use) + ((300 gp x 1 for the spell shield of faith x 9 for the caster level) / 10 for 10 charges per use)).
Note: A cleric can recharge the staff of because talking plants are awesome because of the presence of the shield of faith spell in the staff but must make Use Magic Device skill checks to employ the awaken function. The awaken spell has a 24-hour casting time anyway, so whatever. This staff permits a cleric who has access to 5th-level spells and is willing to commit the resources to being able to use it and recharge it to awaken a plant once every 10 days. You weren't going to use it to cast shield of faith anyway.
The Crown of Bow before Your New Master, Plant Dudes
Aura: Transmutation; CL: 9th; Slot: Head; Price: 36,200 gp; Weight 1 lb.
Description: Once per day on command, this light steel crown depicting moping hermits being evicted from a forest by a valiant metal-armored cleric allows the wearer to spend 24 hours to use the magic of the awaken spell.
Construction Requirements: Craft Wondrous Item, awaken; Cost: 18,100 gp (Math: ((1,800 x 5 for the spell awaken x 9 for the caster level) + (2,000 gp for material components x 50 for an uncharged item)) / 5 for 1 use per day).
Note: The spend 24 hours in the description is not a new requirement but a reminder of the awaken spell's casting time. This has the advantage of being usable by anyone without needing the Use Magic Device skill.