Firstly well done on working through your debt and being 2 months away from clearing it.
Specialist lenders will value this being done, especially when out of monthly income.
I would steer clear from naming specific lenders as one of these lenders may accept a part of your situation, but not something else that you are unaware of. Rather than over thinking and researching lenders and their criteria, it may be prudent to let an experienced broker guide you.
Lastly, you should never really try to manufacture change in your job/career to look to be more attractive to a specialist lender. Zero hour contracts are not always a deal breaker, but what type of role you are, how long have you got a track record for and any gaps in employment history are critical..
Each of the lenders you have mentioned have very detailed criteria and ever moving criteria and product guides and dependent upon the exact details of your credit profile and situation; some may lend sooner than the 2 years from defaults you mention..
Best of luck