I'm not really understanding the problem other than if severance pay + income is above the line for the program you want.
Tax returns are filed and as long as your paystubs / bank statements qualify you under the income rules I don't see a problem... and a gap in employment isn't a deal killer even from the big banks in my experience, just explain it.
Mine was literally: "A buddy and I were working on a new image compression algorithm, and five months into it Google open sourced a project that did what we were trying to do and there wasn't enough of a performance gain in ours to justify using it, so we killed it and I went and got a job." Was all true in my case, but to be clear here I could've been working too.
Ultimately most lenders understand the racket and don't care as long as you can tick off the checkboxes without any can't touch issues: money is made in originating loans, not in denying them, so unless you are clearly broken it should be solvable.