[...] is there an possibility I could find something 2nd hand that will feel and sound more like a real piano at this budget?
"More like a real piano" sure. Maybe not "like a real piano" but close enough.
Some of the really cheap ones would be the various Casio CDP models. If it's 200 series (CDP-230R etc.) it has plenty of sounds and rhytms too. The 100 series was a more simple one.
Casio has the Privia range too i.e. models with PX in the name as in PX-160 etc. Since their 300 series they've had slightly improved key actions from the CDP models and the latter ones (x50 series and onwards) will have a more realistic piano sound too.
Pretty much anything with 88 keys and hammer action is going to "more like a real piano".
Old Yamaha Clavinovas can be okay for a beginner too if they are in a good condition and the seller isn't stuck in the past in terms of the price as in "It cost £2500 brand new 20 years ago, so I'll need to get £2000 for it." Those people do exist...
Some slighly older Kawais can be pretty good too. I have an ES-100 and I'd say it's worth ~£300 or so. (If I were in UK and if were selling it.) Same goes for Rolands and Korgs too: If they aren't really old they may be just fine.
But the problem with used digital pianos is that there are so many various models and it's difficult to tell which are the good ones and what are they really worth.
Then of course e.g. Gear4Music has some really cheap brand new ones. Seem like the SDP-2 has the better piano sound but the SDP-3 only has a real hammer action, so the SDP-2 will feel lighter than a real piano. Which isn't necessarily a problem. It depends on what you want.