Yes - mixing sealants will either make no difference, or both will interfere and neither will work.
Start by checking the SDS (aka the MSDS)
Finish Line: http://www.finishlineusa.com/files/Tubeless%20Tire%20Sealant_EU%20SDS_UK%20English_6%20November%202017.pdf
Both are based on Propylene Glycol with Latex embedded as the solid. So there's similarity in their basic makeup - its not like one is oil and the other is water based. However I still would not mix them; Similar in this case means you don't need to dispose of the new tyres.
- I would unmount both tyres, pour out the existing sealant and dispose safely.
- Then wipe them out with a cloth rag or paper towels. Also wipe down the rims.
- Leave to sit open over night to let anything remaining evaporate.
- Then reassemble your tyres onto the rim and add you preferred sealant brand as per the instructions.
And consider doing this yourself - no need to pay a bike shop for a simple tyre swap.
Based on that, you may choose to try a couple more months on the original sealant, and go through this process after ~5 months because by then it would need a topup anyway.