B737 , I'm sure your relative is well meaning, but I suspect knows virtually about aviation.A type rating with no experience is no help in getting a job.As has been mentioned on this thread, you would have to do the entire conversion course again, and be bonded. The fact that you had a type rating would not necessarily indicate to the airline that you had the aptitude to actually fly the aircraft, for all they know it might have taken you much longer than average to actually pass the course!
If your relative insists on spending this money on you, I'd either ask for a decent car, or possibly help in buying a house when you get the job you want.
It might be worth approaching the low cost operaters and asking to pay for your training in their training system,I'm sure they would love it, but as you are aware by the wording in your post, you won't be doing anybody a favour by going down that route.
With your hours, I feel that you will get another job without too much trouble, it just may not be on a jet type. There is a shortage of experienced pilots. It is true that with the right mix of experience/type rating all you need to do is pick up the phone, but to be in this happy state you need at least 3000-3500 hours, airline experience preferably with command time and at least 1000 hours on the type that your prospective new employer operates.
With your time you fall into the "employable" category, but unless you are very lucky you are more likely to get a turboprop job.
Unfortunately, despite what various flying schools tell people, there are many just-qualified pilots on the market, you don't fall into that category, and are currently flying commercially, which counts for a lot at the interview.
Finally , a word of advice I was given nearly 30 years ago, and has already been mentioned, it is always easier to get a job flying a small aircraft after you've flown a large one than it is to get the job flying the large aircraft. If you really don't enjoy flying a bigger aircraft, go back to piston twins, but you'll lose nothing by trying bigger!