Hmmm, it sounds like you didn't get very good communication from your 1st lender.
The delayed financing loan, allows you to do a cash out refinance within the 1st 6 months that you are on title, with a couple of caveats. You cant have any lien on the property. Basically you had to have either paid cash or like in your case, you used a HELOC from another property.
You can do the Delayed Financing up to 75% on a non-owner occupied property and 80% on an owner occupied property. However you can only get the amount you paid for the home +closing costs rolled into the loan. You cant get any rehab money back out, as its only the acquisition costs + your refinance closing costs as a max. loan amount.
If you want cash out for more than the acquisition costs, you must wait until you have been on title for 6 months, at which point you can have up to 75% Non-Owner or 80% Owner Occupied without the limits of acquisition and closing costs.
These are Fannie Mae loans. The lender that you are working with now, is probably setting you up with a Non-QM or Portfolio loan. On some of these loans, the seasoning time before cash out can be zero waiting time to 3 months. If you are doing a portfolio loan, the interest rate tends to be about 2% higher than a Fannie Mae loan on a 5/1, 7/1, 10/1 ARM.
You're at the age old cross roads of do I wait out the 6 months to get the best rates and terms (Fannie / Freddie) or do I take a higher rate to be able to access the cash within the 1st 6 months, or do I do Delayed Financing and only get the acquisition costs plus refinance closing costs? Its only going to come down to 1 of those 3 choices.
I hope this helps?