From what I've been told, the issue is not the tank it's the septic field. You're not supposed to run animals on top of your field. That said, they were very vague about why. 1. you're adding nitrogen and pathogen load to an area you want cleaning that up from the septic system
2. they think over-crowded typical barn-yard management where the ground tends to get compacted
3. the reason you already gave - pigs can get sick from the same things that make humans sick
If there's enough
of a gap between the end of the field and the area you want to run the pigs, I'd try to plant a hungry sort of plant species as a "buffer" to soak up anything coming off the field +/- a wood-chip trench with mushroom
spawn, and keep an eye on the soil condition in the pig area.