That is a really sensible spec and budget and totally achievable. I must admit that I went for the SE spec as I didn't want the 18" wheels and stiffer M sport suspension. Being a relative noob still, I am not sure if the i spec has sportier suspension than the SE or not (others will be able to advise) but the handling of my SE with the work I've done on it is just great and good enough for my driving which is spirited for sure but I'm not a nutter! - Plus my local roads are in a real state here in Sussex.
I managed to pick up an 04 SE with 95k full history with a great spec including the non-M sport heated electric seats, cruise, reversing sensors, satnav, speaker upgrade in stirling grey for 3k. I have spent 800+ getting it sorted in terms wishbones, brakes and shocks which I have replaced doing the work myself but that was part of the attraction as I wanted a car that I can improve as a project in lockdown. I've also replaced the gearbox and diff oil which I 100% recommend at that mileage and refurbished the leather seats which now look new. (I got a specialist kit to do that which while expensive was worth it IMO). The engine and drive train are so smooth on mine as to bring a smile every time I go out in it and I simply don't want to put it back in the garage, just keep driving it whenever I take it out lol!
One thing I would say is don't worry too about the number of owners mine has had loads BUT with the file and history it has clearly been loved and cherished by each and every of of them. It shouldn't put you off at least taking a look and I wouldn't rule the SE spec but as I said I am not sure of the i vs SE differences and there maybe good reason that you would like the i.
In closing I can honestly say that I have never owned a car that gives more smiles per £ than this one. I had a Jaguar XF before and although that was amazingly comfortable, refined (and quick as well) and I absolutely loved driving around it in, The Z4 is just more fun and engaging.
Good luck with your search and I really hope you end up as happy as I am with my Z4. Cheers Dude, Andy,