A destroyer is simply too expensive to employ for that kind of mission. You're better off with one or two really well optimized frigates. Faction commissions aren't a living on their own, they're more of an excuse to get paid on top of other bounties/loot. If you haven't already, invest two points into getting the 50% maintenance reduction skill. Even if you don't get any other leadership skills its still worth it.
Generally those kind of battles you'll be facing large numbers of crappy degraded combat freighters, so what you want is a flux efficient ship that's good at killing hull and can backpedal decently fast (since its kinda like a zombie movie). Unfortunately the pirates also employ lashers, which is one of the better hulls out there and for a pirate ship of that size is well shielded. Generally I'd say you want either a TT frigate with energy weapons, and figure out some way to kite the little guys (wolf or TT brawler backpedaling is a nice cheap way to get some work done), or you want to go with a lasher of your own. Kit it out with at least 3 autocannons and set them to an autofiring group, then if you come up against a ship without shields turn off the autofiring with ctrl+num. Give it either salamander missiles (for the reasuability/utility) or if you can get an extended missile racks upgrade put the hammer rockets on it. 8 rockets pretty much means 8 dead ships and the low OP cost of the rockets offsets the high OP cost of the upgrade. Pay attention to the faction fleets in the area to infer where the pirates are, and try to killsteal from fights that are already in progress.
You might want to transition away from bounty hunting for now to build yourself a nest egg. Smuggling is good business and your fleet is decently suited for it (except that drone tender is probably too slow). This advice is all based on a previous version of the game, but anyway. Hack into the faction relays to get a constant stream of information about trade events. If a trade event sends a useful or expensive item down in cost, buy it in bulk on the black market and then sell it elsewhere or use it for yourself. If the price goes up from an event, find a cheap seller in a different system, buy on the black market, then ferry it into the system and sell on the black market. As long as events are in progress, you can keep buying and selling at a profit in multiple trips. If you find an event that lets you buy and sell from two markets that are right next to each other you can make a fortune overnight. Don't carry contraband unless you're confident in your abilities, take legit transport contracts to pass the time between your smuggling gigs, and always have one faction that doesn't know you're a criminal.
All that being said, your little fleet could probably grab a few of the big bounties if you could afford the fuel to get out there. Just make sure the bounty doesn't expire before you arrive...