Thank you so much to both of you for taking the time in replying.
I will answer some of your questions, and ask some more!
@geekaMaxima
Is the PI also ok with you going from (presumably) FT on the project to 50% PT?
We have not discussed it yet. This is all in my head for now while I try to work out options!
You are right in that for this project 50% will not be feasible, but the 50/50 split was just an example, I'd say that a 80/20 would be doable and I believe the funders would allow a no-cost extension (as this was discussed for other purposes).
I'm not sure what you mean by a collaboration agreement between the two universities, though.
I was not sure either, someone just mentioned it in passing to me today. What ManInTheMoonMarigold says makes sense.
If you don't mind me asking, why are you so concerned about continuity of service in your current institution?
I might be completely wrong but I thought otherwise I would lose any redundancy pay?!
@ManInTheMoonMarigold
Your PI's new institution could issue a subcontract to your existing institution or the two institutions could agree to some kind of secondment. This depends on the rules of the grant and whether the two institutions are happy to do it.
This would be perfect. Are there any cons? I am wondering why no one (HR/HoD/PI) has mentioned this as an option
For example, some universities will move people with more than 4 years continuous service on at least two successive contracts onto an open ended contract.
Yes, that's my case.
There would be a small increase in redundancy pay,
This will show how naive and uninformed I am, but hey -
if an RA had been on 3 consecutive 3 yr contracts at time s/he was made redundant, is the redundancy pay for 9yr or 3yr? 
I have been at my current institution for >10yr
Potentially, it could mean that there is no application process, just a rolling on to another contract.
That would definitely be the case for my next job.
How about grade and salary - would leaving and coming back not have an effect on this?
Thanks so much again for your help