I know there have been previous forum chats on this ... so worth looking at those.
I don't have a Sport but assuming yours is an S2 then it's the same tyre size and on my Coupé I have Falkens (165/80 R14) which I've been very pleased with. Seem to give all round good performance.
If you wanted a more 'period tyre' then Longstone do a selection but I can't comment on these; other folk may be able to though:
https://www.longstonetyres.co.uk/classic-car-tyres/lancia/fulvia.htmlFor a battery I just opted to use a suitable one from Halfords. The 'HCB063' I have gives plenty of starting oomph and I've had no issues with.