Refinancing with less than 20%
Plenty of lenders will do it. The question is not whether you can, it is whether you should, and the answer turns on one fact: LMI does not transfer. Your existing premium was arranged by your current lender for your current loan. Moving means a new policy and a new premium, paid again, for cover that still protects the lender rather than you.
So the sum is straightforward. Work out the annual saving from the better rate, then compare it with the new premium plus the switching costs. Unless the rate gap is unusually wide, the premium tends to swallow several years of savings, which is longer than most people keep a loan before the next repricing conversation.
The better move while you are close
If you are within a few percentage points of the threshold, the cheapest path is usually to stay put and ask your existing lender to reprice, then refinance later once you are comfortably under 80%. A ten minute review will tell you whether that call is worth making, and it costs nothing either way.