Yes, it's called 'nesting'. Kids stay put, parents come and go. It can work but I don't think it's simple as it sounds.
You and he would in essence be supporting 2 households @ 50% each if you both rented your own places. Could you both afford that? You need to be sure also that he'd keep up his end of that financial bargain. I know of one couple who rented a one bed flat and shared it, each there when the other was with the DC. It did NOT work because, although they'd worked up an 'agreement', the man wouldn't keep to the agreement of 'no shagging here' and he carried on believing that it was his STBXW job to be chief cook and bottle washer. He wouldn't change the sheets as agreed upon and if he did he'd leave them for her to wash. He left the place a mess for her to clean up and although he'd load the dishwasher he didn't soak & scrape burnt on pans or scrape food bits in the bin. He also would never turn it on. I could go on but you get the drift.
If you don't mind my asking, why do you feel you can't be in the same house with him? Is it that the feelings are too 'raw'? Because that will fade in time. I'm not saying what to do, but try to play the long game rather than basing decisions on what you feel right now. If you try 'cohabiting together apart' it may be that he decides to leave because he won't get the full 'bachelor life' whilst in the family home. And he especially may not like having to do 50% of all house and parent duties. Because as of right now anything you do for him needs to stop. No laundry, no 'admin work', no cooking for and no cleaning up after him. If there's another bathroom, start using it and let him clean the one he uses. Sit down and make a list of 'domestic duties and 'invite' him to sit with you and work out who does what.
A divorce won't take 'years' unless he is going to fight you every step of the way, argue about settlement and/or child arrangement, or is going to hide money so your solicitor has to use a forensic accountant to look for it.
Listen, right now who lives where and how long things are going to take are 'tomorrow problems'. See a solicitor. They can answer a lot of your 'how long' questions. As far as who lives where, if he mentions 'reconciliation' tell him you need time alone to think things through. Let him think there's a chance even if you know down to your marrow it isn't happening. That might encourage him to stay at his mum's at least for a bit.
And don't think anything you do or say (within the law of course) to get what you need, be it information for divorce or staying in the house, is 'wrong'. He has violated your trust and his marriage vows. You owe him nothing, he deserves no consideration or honesty from you.