Do I need to update the wiring in my bathroom to replace the gfci receptacle?
Bathrooms frequently have high wattage appliances such has high-wattage hair dryers or space heaters.  And just because YOU don't have one of these, you might eventually have a guest that brings their own.

Plan for success, and comply with the code.  And make sure and get a good name brand GFCI, as there are Chinese knockoff's out there that are known to be bad.  I was having problems with my GFCI's in a new home, and when I pilled one out chd looked up the UL listing code, I discovered that the UL number was fake, and my new home had a half dozen fake gfci's... replacing them with properly certified Leviton's solved my gfci issues...

Of course complying with the code could mean you have to run a new wire, if your existing wire isn't to code.

My rule of thumb is "things are in the code for /good/ reason, and electrical is one area never to skimp on..."  If you skimp, you can cause a fire or death...  Don't do it!

Sure, you may be grandfathered in, so what?  the code is for safety, and many many lives are lost to grandfathered old wiring.  Just having an old outlet can cause a fire.  I'm a firm beliver of periodically replacing outlets that are subjected to heavy loads.  A hair dryer, space heater, microwave, toaster, window unit, etc...  All of these draw maximum current through the plug.  As the plug ages resistance goes up.  The resistance generates heat under max loads.  That heat softend the metal contacts, so that they have less spring pressure, as well as more surface corrosion.  With time the contacts can become bad enough to start getting hot, and eventually starting a fire in the wall, a fire that may take some time to notice until it is too late.

The bathroom outlets are among the ones most subject to current related wear, the next are in the kitchen, and everywhere you plug in the vacuum cleaner, or potentially a space heater.  All of those outlets should be replaced every 10 years or so if they get routine use.  The fact you have old wiring under an old code set is your first clue that your wiring is old enough to need upgrade/replacement for safety.