I have been in this situation, and here would be my 2€:
I'm not sure if this is the right thing for me. I'm not sure if I want to take on that responsibility
"Responsibility" is a weird thing. There are some areas in life where responsibility is real - for example if you are driving a 40t vehicle through a crowded city, you better make sure your driving is responsible; or if you are getting children you definitely feel the weight. But as a team lead? Sure, in business speak people might call you "responsible for the team", but in my experience this kind of responsibility is very, very light-weight in a team as you describe it.
The way you describe it, I find it highly unlikely that you will be fired or in any other way penalized if you fail at your new job; you will, worst case, be demoted back to senior developer, and the experiment will end. You can easily talk about this with your own manager beforehand and make it clear that you will stand up to the challenge if they allow you to grow into it and if they agree to give you a "probationary phase" of a few months, after which you meet again to formally decide if it has been a good idea.
As a team-lead you will at first simply be the spokesperson for the team, i.e., the interface between the "inside" and "outside". This job is basically what your responsiblity will consist of. But you will hardly be carrying a life-or-death weight on your shoulder.
You will need to have very good communication skills to be able to talk about the same set of facts with your managers and with your team members, in a way which appeases both of them, while likely not being able to influence either of them very well. This does not mean that you need to be good at lying or subterfuge, but that you have to understand what your uppers want and bring the important messages through to your team members (in a way they can understand and accept), and the other way around.
All the rest - giving a direction for the team, onboarding new members and such, will come in baby steps and you can decide how to approach these things, i.e. if you want to become a despot or a "primus inter pares" boss. The latter looks like your style, and it is a perfectly fine way to lead a team; that is you do not enforce any technical rules, but you place responsibility for technical topics (i.e., the projects you all work on) on those persons that actually work on them, cleanly separating "team lead work" from "IT work".
or if I am best suited to it,
One way to find out. You will start out with the benefit of the doubt.
but primarily whether or not I want to have to deal with the clashing personalities and the drama.
This is the real job you are expected to do: get rid of, or at least alleviate the drama, and reconcile the clashing personalities. That is, in my book, the part where real team leaders shine, at least in our IT world where most (senior/ish) developers don't need that much of "being led", anyways.
Believe me, if you have a relaxed, friendly, and yet, when needed, tough teamlead, there is not much reason for drama on either end. He will protect the team members from "drama" from above, will take the needs of the team upwards; at the same time you will want to see if your management really needs something from your members and then try to get it out of them in a way that is commensurate with their needs. And so on.
Also, this organization will likely be unable to offer any extra compensation for the adjustment to a lead type position,
They are, though, able to offer the chance to put "team leader" on your resume. As a senior developer you already had responsibility to get your software right, and I'd say that kind of responsibility should be much heavier than that of an internal team lead.
It's not like you are tasked with building a new team from scratch while meeting budget and time deadlines, with your own neck on the line all the time. At least you should probably make sure it's not going to be like that. You have all the arguments on your side; obviously the team is in a bad way right now, so your management cannot punish you if it doesn't work out. Make sure they know and agree to that.
and I doubt they can offer other perks like a lump sum of PTO.
Again, unless you are going to put in a lot more hours per week to fulfill your new role (and you really should not have to - you will need to reduce your dev work appropriately) then I would not worry about that at all.