If it is summer, depending on the severity, probably just use scissors or clippers. At least, that is my solution with cats that manage to get junk in long fur, and where water is a nucatlear bomb with razor appendages.

I can’t say anything relevant about animal health in this area. However, with soldering electronics or pipework, natural flux is pine resin, which is basically hardened sap. It is mildly acidic and has a very low melting temperature. It is water soluble but also tends to leave a sticky residue on any surface once dissolved. When I am etching and assembling circuit boards, I use a lot of flux and must wash it off once I am done. I use 90% alcohol in an ultrasonic cleaning bath. It takes using “dirty” alcohol for one pass to get most of the flux off of the boards, but they come out completely covered in a thin layer of flux. Then I do another bath with fresh alcohol. Once removed they no longer feel sticky to the touch, but they do feel, let’s say “slightly easier to grip.”

If I’m having trouble getting a board clean, I can put a glass dish on my hotplate, heat it to 50-60c and then poor alcohol into it before adding it to my ultrasonic cleaner.

This is intended as an abstract illustrative example of the properties of pine resin. I don’t know how a sufficient based soap will interact with pine resin, but water/alcohol seems to dilute. The dilution will likely move the isolated problem to the skin where the acidic and sticky properties of resin will likely be worse for the animal. Thus my reasoning to clip off any offending spots if possible.