Any idea to convince the mortgage lender to reduce mortgage interest rate?

You can carry required financial documents that can prove that it'll be easier to be current on the mortgage loan if the loan terms are altered.

If you are regular on making the payments, you can get the approval for the loan modification. However, you have to try to best to convince them.

you can shop around for refinancing and talk to your lender about loan modification and analyze which one suits you the best.

Yes.. definitely!!!
What you are asking, is what we call loan modification!
You can completely do that.

You can also refinance your loan if needed.
But if the lender agrees for modification, then go for it!
Trust me it helps a lot!

Opt for refinancing with a new lender. Shop around to find out how much other lenders are charging. Refinance your loan with the lender who is charging the lowest interest rate.

Ofcourse you can do that, it's called loan modification. JUst be regular on your monthly mortgage payments.