Most food banks only allow short-term use, so having families rely on you for two years is unusual and I'd argue, unsustainable. I think you have to ask the question as to why some families are still dependent on you after that time - I suspect some of them are not claiming their full benefits entitlement and/or servicing debt they can't afford.
I think I'd look to partner up with a local CAB-style advice charity and effectively 'borrow' some of their volunteers for a short period of time to see if you can get to the root of the cause. You said one of your trustees was a benefits expert, but given the scale of your problem, one person isn't enough - ask for help. Plus, a benefits expert isn't the same as a debt expert.
There are, for example, CAB volunteers who also volunteer for Crisis seasonally - if you reach out to the right contacts, you should be able to get that extra short term support. Charities work best when they work together.
In the event you work out that some families really can't cope with your food bank long-term, I'd look to see if it was sustainable to run a subsidised shop, i.e. access food through you at a much reduced price, but for some money. It might enable you to keep helping existing service users for longer whilst still being able to help new service users.
I'd also consider scrapping the access to free fresh fruit, veg and protein. Food banks are supposed to be about survival, not luxuries, which is what those are. If you scale back what you give out, you'll be able to help more families survive. You have to pay for fresh fruit, but you could make a local appeal for people to donate tinned fruit. I think you're going to have to reassess how you make use of your funds to have the widest impact.
Unfortunately, with the cost of living crisis, you're looking at either supporting some families so they eat well, or supporting more families so they eat well enough.
PS Well done to you and the others. Food banks aren't just physically demanding to volunteer in, they're mentally tough work. You're doing a good thing.