I had a solution. One thing I briefly considered was trying to dump more priming sugar (as a solution) into the pig through the spout but that seemed like a difficult task, even using tubing to funnel it into the right place. My primary fear was that even if I could successfully get sanitary priming solution into the Party Pig that it would overcarbonate and explode. Quite a dangerous mess. Instead, I found out that Quion, the company that manufactures the Party Pig sells a CO2 charger that is intended to replace the handpump sold with the Party Pig that you use to activate the pressure pouch. (If you've ever tried to use that handpump then you know it is incredibly difficult to use.) I figured if CO2 could go in after filling it could also be used to repressurize the Party Pig after some of the beer was dispensed. I asked the folks at Quion if they had any thoughts about repurposing the CO2 charger. The thought had never occurred to them but they didn't see a problem as long as I didn't overpressuring the Party Pig.
In September I filled one of my Party Pigs with my so-so ESB and tapped the Party Pig-turned-cask in my fridge early October. After drinking on it for about three weeks I had about a gallon left and the pressure had equalized, resulting in the typical slow pour. I removed it from the fridge, stood it upright, shoved the charger in the spout, hit the dispensing button and shot in some CO2. I pressed against the Party Pig to see when it started to get firm and stopped the CO2 once it became firm. Tossed it back in the fridge and checked on it a week later. Good news: not only did it pour quickly and slightly foamy but it had a cleaner taste than the last time I had tasted it. I don't know what happened but by the third week or so the beer started taking on sort of a rough character. Maybe it was oxidation. What I had after the CO2 was fruity and delicious.
The bad news is that it only seems to pour 1-2 pints before the pressure gives out. It's not that big of a deal to shoot more CO2 in after a pour or two but it will definitely require an increasing amount of CO2 the more I drink. Food grade CO2 cartridges aren't terribly expensive (although not as cheap as refilling tanks) but I'm sure if I drank the cask faster I'd have to use the CO2 charger less frequently.