You don’t sound keen on the larger house that needs work, and you should choose whichever suits you all best...but in your shoes I’d go for the larger house.
The smaller house won’t be where you live forever. And after 5 years (probably less!) you’ll have had to redecorate hallway, kids rooms, after 8 years the living room, after 10 years a new kitchen.... then add up conveyencing, LBTT, EA fees (when you sell), and compare that to the cost of doing the fixer-upper.
However... might be as well living with the kitchen and bathroom and doing up the rest of the house properly (which won’t cost anything like £10k unless it needs significant repairs) If you start with living room then youhave somewhere lovely to relax even if the bathroom is avocado green!
Bedrooms are generally easy to do, assuming you just paint and a new carpet they’d not be expensive. If you reckon on £300 per room (paint and carpet, very rough guesstimate without knowing sizes), and £500 for living room then round it all up to say £2k just-in-case. Doesn’t include furniture though, not big costs like sorting damp, replacing windows, wiring, heating, etc!
Bathroom and kitchen will be substantially more, and unlikely you could do them to a robust standard (am looking at you, DC!) and hallway for the remaining £8k.