One mistake I see people make with SBA loans is they look to what the SBA will allow and not what the actual lender will do. In most cases, the difference is quite large.
Meaning, just because the SBA will allow up to 90% financing on 7a loans does not mean the bank is obligated to do so. I'm seeing most banks lending in the 65-80% range for the 7a program range with an occasional loan for 90%.
Since you have a rehab scenario you are likely going to have to come to the table with a lot more than 10%. My SBA lenders don't mind funding minor improvements, working capital, and/or equipment as part of a loan, but I don't know of any who would embrace major rehab work right now.
If this is a start up business you really need to talk to a SBA lender who has PLP (Preferred Lender Program) status to discuss their appetite for start up businesses and what is a realistic loan if they are lending on them.
From the collateral standpoint, the SBA program is very flexible. While they prefer to use other income producing collateral, they are not opposed to attaching the equity in your home, and will actually allow co-signers who are willing to pledge the needed collateral.
As far as when you would inject the cash, it would be like any normal closing in that you would bring it to the closing table.
I don't mean to be the bearer of bad news so please don't take this post as me trying to rain on your parade. It's important to talk to an actual lender to get an idea of what is possible so you can decide how to proceed based on real numbers.