Per Crawford, No: you can't use the paladin's Divine Smite with a thrown weapon.
The problem here, as you have correctly identified, is that the language is ambiguous: does "melee weapon attack" mean "a melee attack with a weapon" (your option 1), or "an attack with a melee weapon" (your option 2)?
Helpfully, rules designer Jeremy Crawford has answered this exact question in a tweet from August 2015 (which is not an official ruling), and in the following Sage Advice Compendium question on p10 of the v1.14 PDF (which is an official ruling):
What does “melee weapon attack” mean: a melee attack with a weapon or an attack with a melee weapon?
It means a melee attack with a weapon. Similarly, “ranged weapon attack” means a ranged attack with a weapon. Some attacks count as a melee or ranged weapon attack even if a weapon isn’t involved, as specified in the text of those attacks. For example, an unarmed strike counts as a melee weapon attack, even though the attacker’s body isn’t considered a weapon.
As such, we should interpret Divine Smite's trigger to mean "...when you hit a creature with a melee attack with a weapon...", which means that you can't use Divine Smite with a thrown weapon, as it isn't a melee attack.
The best way that I can see you improving your combat effectiveness is through the specific spells you choose.
As you had said, money is sort of tight in your campaign, so you need to make due with what you have at your disposal naturally. Luckily, this isn't hard considering that you have a +4 Cha score (Giving you anywhere between 1-4 additional spells per day at your current bonus). So considering your want to improve how well you do in combat, focus on preparing combat buff spells
These are the spells that I think you should prioritize (keep in mind that I only have the core rule book on hand):
+Bless Weapon(1st): A very situational buff, but powerful when used against the right enemies (ya did good on this one)
+Magic Weapon(1st): Prepare this instead of Bless. Bless may give you a bonus against fear, but Magic Weapon increases your damage as well as your attack bonus.
+Bull's Strength(2nd): I know that you put most of your stuff into Ranged Weapons, but everything on the list available to me is either a buff you don't need or something that seams too underhanded for a "rock star" (also, I don't think you need a spell that lets you take half of the damage that your friends take when there is a fully-functioning tank). As you may already know, Bull's Strength increases your Strength Ability by 4, so at the very least, you increase your Combat Maneuver Scores. This means it's easier to absolutely wreak an enemy's day by affecting how they move, inflicting statuses on them, or simply moving through them.
+Greater Magic Weapon(3rd): THIS is one of the key spells you should prepare when you get into the upper echelons. It's like Magic Weapon, but it gives you additional bonus amounts for every four levels your character has. Also, this spell specifically states that you can cast this spell on up to 50 arrows in the same quiver; this means that if you have to switch to another bow, or if your party also invests in bows, the spell is used to its maximum potential between multiple weapons.
+Prayer(3rd): Anyone in a 40ft radius of you receives a +/-1 to every roll depending on their loyalties; That's an increased advantage of 2 against whatever enemies you are fighting!
+Dispel Chaos/Evil(4th): these two spells give you +4 against the attacks of the designated alignment and automatically dispels effects cast by said creatures or spells that are of that alignment
This last one is just in case you attempt to build physically or your Fighter is being cornered by a Demon or something...
-Holy Sword: PROS: weapon automatically becomes +5 AND deals an additional 2d6 damage to Evil Creatures. CONS: Can only be applied to melee weapons and negates the powers of whatever weapon it is used on (that last bit could be fun to use against an enemy using an all-powerful cursed weapon, but again, this is a bit situational)
I hope this helps.
P.S.: Your party should invest in Heavy Repeating Crossbows once you hit level 10
My Experience with Ranged Smiting
I have created a magic item for my players that allows for ranged smiting. They are magical arrows that cost gold (I give my PCs the ability to craft magic items), and they deal smite damage on hit. In exchange for ranged smiting, however, the paladin must declare the smite and expend the spell slot for the smite before they fire the arrow. If the attack misses, the smite is wasted.
Why I like this compromise
I feel like this creates a useful middle ground for the paladin. In play, she still goes for the melee attacks most of the time; only when the enemies are truly out of reach does she turn to the arrows. In effect, you are balancing the incredible versatility of a ranged smite with the possibility of wasting a spell slot, turning the smite into a ranged attack roll. Moreover, because firing the arrow requires DEX, your paladin has to make the same spellcasting stat vs. attack stat decision that they normally do. Finally, because they are consumable magic items, you can give out fewer of them if you think they're too strong, and there's an aspect of resource management to them as well.
Also, this makes sense in-universe: the Paladin can choose to use the smite from her weapon because she's always holding it, but if she shoots an arrow, she can only infuse it with the smite while it's in her possession.
How this applies to your situation
Unlike your situation, my party was not all paladins. However, it's clear that even within your party, there are distinct specializations, and not all paladins will be able to benefit from ranged smiting. A dex-focused paladin would benefit far more than a str-focused paladin, for example, and it might feel unfair if the ranged paladin can do everything the melee ones can, but be out of harm's way as well.
Since your situation is unusual, I'd suggest phasing in smite arrows slowly. At first, you can introduce arrows that can only deliver 1st level smites. If those work out with the party, you can introduce arrows that are capable of higher level smites, as well as the smite spells. Ranged smites too powerful? Then give fewer smiting arrows. The ranged paladin's too weak? Then give more powerful arrows. Because the arrows are consumable magic items, you can easily "titrate" the ranged smite ability you give.