My Wilderlands PCs may spend several days or a week or a couple weeks at a time adventuring. Maybe they get into 1-2 fights in that time. They do lots of other stuff, like trekking across rugged hills, navigating the endless ocean to mysterious isles, flying winged steeds over high mountains, et al. And of course lots of meeting people and talking with them, discovering ancient relics of lost aeons, delving hidden mysteries of the forgotten past, etc.
Exactly. The gritty realism rest variant is absolutely perfect for your game. PCs must spend an entire week going nothing but chilling out in town in order to have a long rest. Short rests are overnight affairs (or even 5 minute affairs).
5E DnD is balanced around 6-8 encounters per long rest, and no more than 2-3 encounters per short rest.
As in - classes are balanced around this assumption, as is encounter difficulty and challenge ratings.
I hazard a guess that [Fighters, Warlocks and Monks] in your campaign are under-powered next to [full casters, Barbarians and Paladins], and appropriately budgeted [medium-hard] or even [hard-deadly] encounters are getting steamrolled.
Instead of mucking about with classes, muck about with rests.
Presuming your PCs spend a month away from home, doing fantastic crap etc, rarely get more than a dozen or so encounters in that time away from home, and rarely get more than 3 in any one single day, implementing [long rest] as = [week long rests] and [short rests] = [overnight] is perfect for you.
Heck; if your campaign only averages less than 6 encounters per month of game time, and never more than 3 in a single day, just have long rests happen in the background, but the benefits only kick in at the start of each game month. Short rest benefits kick in overnight. Players are healing, studying spell books, resting etc every night, but it takes a month for any substantial benefit (a long rest) to kick in.
Bang presto. Paladins and Barbarians have to ration smites and rages over a whole month (of 6-8 encounters). Wizards cant solve every problem with a click of their fingers. Fighters only have to ration their crap over a single day (of 0-3 encounters). It takes a month to heal from injuries, and your hit points + hit dice are expected to last you 4 whole weeks.
If your campaign runs at the pace you claim, then that should be around 1 long rest every 6-8 encounters, and a short rest every 1-3 encounters.
Classes will balance at the spot they were designed to balance, and you can stop ramping up encounter difficulty (which only encourages nova strikes and nova tactics, devalues player choice and abilities, stops fights being boring games of rocket tag, makes the chances of a campaign smashing TPK more likely, and nerfs certain classes like Fighters, Monks and Warlocks unfairly).