I don't think that compression and flare fittings use the same thread size. (I could be wrong there though.) You might be able to find a compression to NPT and then go NPT to PTC. Or you may find compression to PTC. Depends a bit on how your line is currently run and accessible. You could potentially cut in a new line easily somewhere else if the branch line is accessible, like a basement. Also depends on what material it is. Pex would be easy, copper not too bad with a sharkbite or other PTC for copper. 1/2" NPT is a common denominator at times or some other NPT size. Fishing anything through is going to depend on the path as you mentioned and then what size holes may have been used for the braided line. Pex is flexible but sometimes difficult for more than two holes, like having to drop down a bay with a fire break in the way. I used to curse those things until I learned they were for safety! A cone shape over the pex tubing end helps to pull it through the holes.
Just an anecdote. When I replaced my water supply lines, I went to pex. I noticed an off flavor for the water coming out of the fridge. The water tasted different on the new supply line that was pex to the fridge supply line. The taste was different than the older pex line to the kitchen faucet. I felt it was the result of water sitting in the pex line perhaps and perhaps that the line was new. I switched it to copper off the main pex line and the flavor went away. It may have eventually went away. I think it would have but I wasn't going to drink it for however long it took! It's a common material for supply lines so I think it was just the low volume of water initially.
