Mike,
Im working that exact situation In VA right now. I am to close on my new property that is being built on 30 sept. Ill describe what I went through below.
In my case I had not used my complete VA coverage in my initial home purchase. In Norfolk, VA Veterans get a total of 458800 in total entitlement. so lets do the math.
The VA guarantees .25 of the loan. = 244,000 (price of my current home) 244,000 x .25 = 61000.
tottal entitlement is 458800x.25 is 114700. Then 114700-61000 is 53800.
53800 is my remaining entitlement toward second property.53800x4 = 2148000. This now is how much house I can afford with having to make no down payment.
Since in my case the new home purchase price is 260000 the VA mandate is that I cover the remainder which is 260000-214000=45200. my down payment is .25 of the difference. 45200x.25=113000. This amount I have to take with me at closing. Again, this is only because I went over my entitlement.
Most lenders I that I encountered wanted to know that the original property was income producing prior to financing and taking the risk on the second one. They asked for the rental agreement and deposit check (copies) as part of the application process. This has to do with you DTI and the loan program That you qualify for.
All things considered in your situation. I would recommend refinance your Hawaii house under the VA IRRRL program. This is a quick process. Most can get it done in under a month. No income verification required.
Take the cash out option, then use the cash as a down payment on Jersey property. 175k should be able to get you about 800 thousand at 20 percent down. The math is not exact on that. Im originally from NY. 800k should be able to put a roof over you head in NJ.
There will arise the question of occupancy. The VA does not care about the rental as long as you have lived in it for some period of time. This period is not specified. Occupancy requirements for the IRRRL is the you have "previously lived" at the address refinanced. Brokers unfamiliar with the program will try to tell you other wise.
Begin by calling your respective VA field office. While you do that, ask for your certificate of eligibility. Have your ducks in a row prior to calling the bank. some will trying to give you crazy numbers. Know what you are talking about before calling the broker.
In my case I had to explain the whole concept a few times before he got hold of it. A few flat out said it was impossible. You can make it happen. It will take patience. The VA office will explain it all to you.