Hmm. OK. But you need a deadline, and you need to "motivate" him i.e. scare the arse off him with what will happen if he can't find somewhere he's happy with.
So, for example, if he would baulk at the idea of losing money to rent a house that's a) large enough to be a step up from the flat and b) imperfect anyway, then you need to point out forcefully that if he hasn't found somewhere he's happy to make an offer on by December 1st, you will be visiting letting agents with a view to renting a suitably expensive/large/practical property to rent. And he'll be paying removal costs twice (and adding the packing service into the deal twice, because you won't be doing it). You don't say how old DC1 is, too, but you need to consider schools and application deadlines too.
You can't buy somewhere just because you don't want to rent, if renting is more practical really. A mortgage IS a huge commitment and you need to be on the same page about the house you end up buying. So you can't force him into an agreement, really, just because you'd prefer not to rent.
Wrt the garden: who will be responsible for it, or who cares most? If mostly you, then you get veto. If mostly DH, then he gets veto. If it will require lots of money to look after from joint finances (in the case of too large, say), then it is a no.
Wrt the house e.g. bedroom size: all bedrooms too small? Objectively, i.e. can't fit furniture and storage in, or just smaller than you'd like? I would prioritise living space over sleeping space, so you need to know what yours and his priorities are.
Location: main roads are pretty much a straight-up no if one of you thinks so, IME. Not sure what the issue with "being by a lane" could be, though.
I do sympathise, OP.