I agree with what clunckdriver posted and other posters that posted in kind.
Before I retired I was a chief pilot and while hiring someone with a type rating was desirable in the aircraft we operated, Sabre 65, Falcon 50/50EX, Falcon 900EX, I never mandated that anyone I hired must be type rated in our aircraft.
In fact two pilots I hired were not type rated in the Falcons, we sent them to school and we paid for the school.
I've never agreed with having any pilot paying for their own type rating to get a job with a company.
That includes Southwest Airlines.
So I would recommend keeping your present job and keep looking.
Oh, and yes, a type rating in a IAJet (Westwinds) is not the most desirable rating to have these days. Don’t get me wrong, I flew a Westwind II for three years and it does a good job, doesn’t hand fly worth a darn, but it does the job.