Purchasing a home in the current environment

So a great house has become available and the wife and I are wanting to put in an offer. I want to give you my timeline so you can help me determine the optimal path forward.

We see this next house as our second to last house as we intend on building our own around 12-15 years from now. Love the prospective house but don’t currently have enough saved up unencumbered to do the full 20% down. I’m not willing to free up the money that is otherwise earning between 8-15% annually or take a tax hit to get to the 20%.

I’m thinking put the minimal amount required for the down payment with the expectation of refinancing in the next 2-4 years. Obviously the rates would need to go down at least 1% from current market for the refinancing to make sense.

I’m thinking I go 30 yr fixed. I know some lenders are offering to lower the rate by 1% for a year to get people to buy now but that’s a gimmick. I know you can recast loans but I don’t want to do that either.

My neighbor was able to effectively refinance his loan without zero costs but is his term was extended to the original term length in order to do so. This was during 2020 and he say the lender reached out to him for that deal. This is the option I would like to have but I have no idea what it’s called (terminology) or whether it’s something that can even be offered as an option up front. Basically, I want to avoid the costs the refinance in the near term, get the benefit of the appreciation in the property to rid myself of pmi, and lower my payments all within the next 4 years.

Do I find a lender willing to do this or is this something I can express to a broker who can find this deal for me?