The problem I ran into with SharkBit was trying to install an outside faucet. A copper pipe worked great but with SharkBite I had to secure the faucet to the wall because if would just spin in place.
There is a SharkBite fitting for that. A professional plumber gave me a tip I've used with success. When using these fittings with PEX and other plastics take a pair of needle nose pliers and yank out the plastic sleeve in the end of the fittings. This prevents the pipe or tubing from misalignment and not seating fully. Removal has no affect on sealing.
I think the tubing from the water filter is likely to be polyethylene tubing. How sure are we that SharkBite fittings are listed for use with polyethylene tubing? I could not find a mention of polyethylene on the SharkBite web site.
I think the best bet is to use polyethylene tubing for the run as long as the pipe will be exposed.