Mixing tire sealants

Yes - mixing sealants will either make no difference, or both will interfere and neither will work.

Start by checking the SDS (aka the MSDS)

Stans: https://www.notubes.com/pub/media/wysiwyg/GIS-004_R3_Stans_Tire_Sealant_SDS_GHS.pdf

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.

1. I would unmount both tyres, pour out the existing sealant and dispose safely.
2. Then wipe them out with a cloth rag or paper towels. Also wipe down the rims.
3. Leave to sit open over night to let anything remaining evaporate.
4. 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.