"By the time we've paid for all that (£1-2k just for surveyors, not inc solicitors)"
I would suggest maybe go back to the Surveyor you've already spoken to and ask about how their costs would be apportioned in this specific situation where both parties will benefit from the work to the wall.
I think it is likely that the Surveyor will say that their costs are to be split equally.
So if the total Surveyor costs are £1,500 (midpoint of £1-2k) then that is £750 each on top of the £3.5k cost of the building work, or £4.25k each.
Now supposing that you then get the work done and your neighbour refuses to pay their share of the cost.
It would then be a case of you going to the small claims court (or, rather, the "small claims track"). In this situation, for a relatively simple case where you have a Party Wall Award and the other person hasn't paid you won't really need a solicitor, this is something you can do yourself.
The cost to bring a money claim of between £3k and £5k is £205 and you get this back from the other party if you are successful. You can also claim very limited costs such as taking a day off work for the hearing (£95) and travel costs.
So, it all depends on how things are with your neighbours. You pay the full amount and just suck it up in order to be friendly with your neighbours or press ahead and ask them for a proportion of the total cost that is acceptable to you.
If they won't agree to a proportion that is acceptable to you and you don't want to suck it up and pay the full amount yourselves it is then going down the Party Wall route.
If you pay the full amount yourself then you say you're looking at £7k. If you go the Party Wall route then it is likely going to cost you £4.25k by the time everything's finished.
But to get back to your original question:
"But any tips on how I could try to go about it so maybe somewhere down the line they might pay us back..??"
Unless you get this in writing then anything they say will be pretty hard to enforce.
You simply need to think about what the minimum figure is going to be that they contribute that you are prepared to accept. Once you've worked this out then speak to them about the total cost of the work and show them the quote.
Obviously, start off with them contributing half and see how it goes. If they aren't open to that then that's when you have to start discussing about how much they are willing to pay.
If it's enough that you'd rather them pay that amount than having to go through the Party Wall procedure then get them to agree in writing and you're good to go.