With 9% second, it sounds like a fixed rate loan, but you should confirm that.
The time that you expect to keep the loan is important. The lowest cost option, currently, is probably the way to go if you intend this loan to be short term.
A one loan option with MI may be more expensive at the moment, but it is a safer long term solution. To eliminate the second later would require a refinance, and of course rates cannot be predicted for that time, but the trends indicate we will have higher rates for some time to come. The refi would also include fees. Mortgage Insurance can be canceled once your loan balance hits a certain point (your lender could give you the specifics).
Options if you are eligible:
VA: WIth 10% down on a VA loan, the funding fee is reduced and there is no Monthly MI
FHA : There will be MI with an FHA loan and usually a 90% conventional loan is similar in payment to a 90% FHA loan, but the FHA product would allow you to put less down if you chose and keep some in your emergency fund.
Your question should be addressed by the mortgage provider that is willing to do the 80/10/10 loan. A good faith estimate should be provided as well as the pros and cons. Surely your question was proposed to the lender. Is it that you don't believe their response or is it that they don't have one.
If you need more information you can always contact Ruth Falcone at Trident Mortgage.
Ruth H. Falcone
305 2nd Avenue, Suite 210
Collegeville, PA 19426
Phone (610) 831-8112