How would you go about addressing these issues?
Well, as I see it you don't need a lot of punch to take down the odd pirate, you just need to be able to get there in time, which means being fast, and you need to be close enough to get there, which means numbers. But you'll also want them tough, if you're losing a bunch of ships that gives the "let's go the mygeen death cloud with a stick and see what it does" types ammo. So you'll want a force of small, fast ships, built in bulk so you can get as many as you can out there. So something like a Saber class or militarized Nova (I'd prefer the latter, as it's the more traditionally federation design of the two). It would still have enough punch to engage peer threats, but as compared to something like the defiant that's heavily optimized for sublight manners and firepower, this one is but for FTL speed and durability first, with firepower a secondary concern.
However, both those ships aren't really intended for long term operations, they have high speed but limited operational endurance, and you can't just base them out of starbases because if ships located at starbases were close enough to handle anti-piracy duties, then there wouldn't be pirates. So instead, you'll want so sort of mothership they can return to to rest, resupply, rearm, and repair. MY first thought was a galaxy class, but those are really expensive, and even if you're assigning a dozen or so Nova to each one, they're kinda overall. So I'd suggest a variant on a slightly smaller ship, like a Nebula, outfitted just for heavy combat and fleet support.
The Nebula is also there to address point 3. The vast majority of the time the fleet is just going to be running around fighting pirates, so there's inevitably going to be pressure to scrap the larger warfleet operations and just focus on pirates, so the dedicated warships needed for role 2 can be retasked to go poke at space things with a stick. So it's important that those heavier ships be tied in the patrol and pirate fighting and can't be eliminated. But they're also heavily armed, for fleet battles. If you can manage to assign each one a few escorts, something cheap like a steamrunner, great, but that will depend on those steamrunners seeing use, if you don't have to have them back up the Nova's to take out pirate bases and the like on a regular basis, you might lose them.
In those rare circumstances where a major fleet battle is coming, this federation military force can't handle it solo, you're just not going to get the funding to build hundred and hundreds of ship of the line. You might have a hundred or so of the Nebulas, maybe some escorts, and then like 500 Novas, but the Novas aren't really intended for this, they're basically BoP style raiders at heart, built to take out the same and win. You'll be worn down to nothing in a short time if you try to fight solo. So instead, what you do with this force is use it as the tip of the spear/force multiplier in a mixed starfleet force, rather than trying to fight the war yourself.