Get a buy-to-let mortgage from another lender. Once that's done, arrange to port the current residential mortgage to the new property.
I asked [Santander] if I could get a new BTL mortgage product for my current house to continue renting it out, then move my current mortgage onto a new house. They said no, I have to sell the house.
It's possible that Santander said that because they don't generally offer BTL mortgages. Start by shopping around.
Because of the potential complications of dealing with two mortgages and two banks, you may want to use a mortgage broker. I've had good experiences with Charcol, but others are available. Even if you pick your own mortgage, it's useful to have someone else to harrass the banks for you. Note that brokers do charge for their services.
Once you have actually applied for the BTL and have received the mortgage offer, then contact Santander to port your existing mortgage. Although in theory you will be moving an existing mortgage, in practise it's treated as if you were applying for a brand new one (albeit with the old terms and rate), so you'll have to go through the whole process again.
The important thing with regard to the BTL is to show that your rental property is not a liability. So long as, with the BTL mortgage, it's not going to make a loss, Santander are likely to ignore it altogether.
If that all works, you then arrange with your solicitor for completion on the two properties to happen simultaneously.
Re "keeping it in the family": note that some BTL mortgages specifically prohibit you from letting your property to family members.
Source: me! I've previously done exactly what I describe above.
EDIT: Regarding potentially selling or gifting to family: if doing this for a number that is well below market rate, there could be potential tax implications; you would need to consult an accountant and/or solicitor.
There are other options; for example, if you wanted a family member to share responsibility for the BTL mortgage, you could change ownership of the property so that you own it as tenants in common with unequal shares, with your relative owning a nominal share (e.g. 1%, or £1). Again, this is something to discuss with your solicitor before applying for the BTL.