It really depends. for a cleric, they are functionally identical.
Here are the damage expressions:
Two hander: 2d6 + str
TWF: 1d6 + str, 1d6
Literally exactly the same when you factor in that your to-hit is going to be the exact same number for both. You only get your Divine strike (I think War domain gets that at 8 and 14 like the Life cleric does) on the first hit so that does favor TWF slightly at 8 and more at 14.
The next thing to look at is the action economy. TWF uses your bonus action every round. Whereas GWF leaves it free. You've got some bonus action spells you might want to use.
Lastly, there is feat selection. A melee cleric is MAD (Multiple Ability Dependent) so you might not want to actually pick up a feat and let both your stats run up to 20, but if you could settle for an 18 str, you might want to pick up the feat related to your fighting style. The GWF feat gives you extra attacks, but the TWF feat lets you wield one handed weapons rather than only light weapons and boosts your AC a bit.
Really, looking at the numbers, the decision is entirely up to you and how you want to play the character, they are quite similar. I give the damage edge to TWF, and the action economy edge to GWF. Ultimately though, at high levels, your cantrips might end up doing more damage than your weapon attacks anyways.
Answering the question as posed in your title (how to run a game for three players without a healer), you have two main options:
Offload the mentally-taxing parts of running a GMPC
I've played in three groups that didn't have a PC healer. In two of them, the DM assigned an NPC companion character to one of the players. The DM did the majority of the RP for the NPC (e.g., speaking, making out-of-combat decisions, etc), but during combat the assigned player managed the character. This works because combat tends to be the most DM-resource-intensive part of D&D/Pathfinder, and you want your attention as DM focused on that rather than on the NPC.
It sounds like this is your first time DMing, so I'll pass along a general tip: DMing is incredibly mentally taxing. Fun and exciting, yes, but exhausting. One of the many reasons GMPCs are not recommended is because, frankly, the DM already has enough on her plate and adding the responsibility of a PC just makes things more difficult for her.
That said, since you're planning to play at least one GMPC, healer or no: look for ways to offload as much of the thinky bits of playing the character as possible. Reduce the healer's stat blocks to just a healer; I can't remember what the class name is right now but there's a 3.5 healer class that literally only does healing. (Its name might actually just be "Healer".) That's the class one of the NPC companion characters used in my example above, and it made things simple because all his choices were "Who to heal and by how much", rather than tactical things like "heal vs attack vs tank vs control". This reduced the mental load on the DM, plus made it easy for one of the group's other players to take over the NPC in combat.
Use items and skills to make up the difference
The other way to handle this situation is to simply ensure that the players have sufficient Use Magic Device, minor spellcasting, and items/cash that they don't need an explicit healer. This is what we did in the third group I played in without a healer. We were a multiclassed bard/fighter (got some small heals from the bard half, plus UMD); a ranger (also some minor spellcasting, plus general toughness); and a rogue (craploads of UMD, wands, and scrolls). Our DM made sure to drop lots of healing potions and wands as treasure, and to have lots of places where we could also buy healing items. We didn't need a healer because our own minor spellcasting and UMD abilities worked just fine.
Looking at your group's breakdown, you could likely do something similar, since you've got quite a bit of spellcasting/UMD power. Just make sure, as DM, that you drop lots of healing items as treasure. The players will figure it out from there.
Best Answer
In short, let them play the same class, but check that they play different characters.
It's not a class problem
You mention that you were worried that they might compete for the same role; this is something to look for and you know that already. But this is not so much a class problem as it is a player problem. For example, if two players want to be the face, it doesn't matter if they are both bards or if one of them is a bard and the other a sorcerer. It matters that both players are trying to do the same thing at the same time.
What you should look for while they create their characters
Talk to them and check these things:
For those first two points, look at their backgrounds and what they tell you about their characters.
That their specialisation is different. One bard might prefer to be a storyteller or mediator while the other focuses on music. One might favor diplomacy while the other favors performance and sleight of hand.
For this point, look at their skill proficiencies, stats and spell selection.
Make sure they have mechanically different roles. In combat, they would ideally have two different battle plans. One might want to kill while the other wants to facilitate kills or protect his allies.
For this, check their spell selection and weapon of choice.
Redundant classes is rarely an issue by itself. Especially in 5e, where party (class) balance is less important than it used to be, especially with a class as versatile as the bard. Even those tools that they will have in common work well together; more bardic inspiration just means more opportunities for skill challenges. Being charismatic just means that your social challenges can be more diverse.
In case they play the same character
And finally, in case they play the same character, make sure they both know about it and they know what they are getting into. You might find the players will play off of each other's role play and become in-universe friends. Or, they will incorporate the rivalry into their role play. Again, the players need to know what they are getting into.
If they do make similar characters, check up on them after a few sessions. If one of the players feels bad about the situation, you would do better to change the characters before the tension seeps too far into the game.
Personal note
To give an idea of how many similar characters can coexist: in a game I am currently in, admittedly a mostly social and investigative one so far, we have four face characters. Two of us specialize in Intimidate/Bluff, the other two specializing in Diplomacy and Sweet-talking. What happens is that one character takes the lead while the other one helps on the roll or goes for secondary tasks.
For example: an Intimidator might try to get info out of a witness by force while the second character tries a Bluff to pressure the witness. Vastly different backgrounds in the two helps for that. One of the characters is ex-military while the other is a quick-witted rich kid.