Without knowing what you have remaining on your mortgage, it is impossible to know whether the payment would be higher or lower than your current $1,200/month.
A 15-year loan is always going to mean you pay less in interest over the course of the loan than when compared to a longer term loan, particularly a 30-year loan. Whether that is "better" or not depends on how you look at it. From an interest paid standpoint, yes, it will always be "better" in the sense that you will pay less interest than with a longer loan (assuming relatively similar interest rates). But if it results in a higher payment that you end up not being able to make and you get foreclosed on, then probably not.
And as previously stated, you will need to make sure that your equity represents 20% or more of the value of the property to get rid of PMI in a refinance (my guess is that it would on a manufactured home, but it is still something to note and confirm prior to refinancing).
My best suggestion would be to contact a lender or mortgage broker, tell them the current terms you have on your loan, and have them run numbers on whether it will be beneficial to switch to a 15-year loan and whether you will be able to get rid of PMI in a refinance.
Best of luck.