Having been through this with a similar lazy and checked out exH you wont be able to prove he hasn't been a SAHD. Even if you could the court wont want to dissect a relationship. As you stayed married in the court's eyes you accepted it. He will argue both roles were equal.
You can decide not to throw more money away. The quicker you get out the more money you will keep in future. My ex dragged it out and by the time the process finished I had paid down alot more mortgage etc. Every month you pay off the mortgage you are gifting him at least half. Ask yourself if you are happy to fund his retirement hobbies / lifestyle from just your pension as that is clearly his plan. There's no benefit to you in reducing the mortgage now or putting more money into shared assets.
The court will take into account his earning potential not just his actual earnings. He will need to explain why he has no tax returns if he's not declaring the money he does earn. Anything that proves he is dishonest is helpful to you. That he has zero mortgage capacity is a problem for you.
If the assets are enough to meet needs and there is some left you have a better claim to the extra. Also if you had assets before the marriage.
He will probably decide he does want a house for him and dc once he realises thats the way to prove a 'need' for a larger share.
The CM will be neligible if he's basically unemployed or on min wage.
I assume you will buy him out as he has no assets. You can ask the agents to value it for a divorce - make sure they know you want a value of what it would definitely sell for within 6 weeks not an aspirational value.
You can rebuild. What you earn or pay into a pension once this is done is 100% yours.
Its really hard to accept you've been financially taken advantage of, but the dc will know which parent was there for them and which was selfish.