Every single prep school will apply their own criteria so it's impossible to say whether any one factor would make you ineligible, or even sway their decision.
I'm sure your accommodation situation is not unheard of, the important thing is that you make all the facts clear on the form; is there a space for additional information that you can use for a longer explanation? Be sure to include copies (or originals if asked) of supporting documentation, even if not specifically requested.
What they will primarily want to assess is whether it would be IMPOSSIBLE for your child to attend if they do not offer financial help. So they'll use your financial information; income, assets, outgoings, to make a judgement on that.
Most schools would expect parents to maximize their income and, if necessary, make some sacrifices to pay fees - but only within reason! People awarded significant bursaries (which are rare at prep level) are unlikely to have much, or any, flexibility in their financial circumstances.
But it can depend on so many other things as well. How much money the school has, how many candidates there are, how your child does in entrance exams, whether they impress at interviews without appearing over-prepared ...
Do make sure the school is well established, financially secure and can actually offer the level of bursary you require. (Not all live up to the claims in their prospectus.) And have a back-up plan!
Good luck, hope it goes well for you.