Should I install a second septic tank or add a long run for an outbuilding?
So much of this depends upon local regulations and soil conditions on your site.
If you decided to do a separate septic tank, you'd also need another drain field (in some parts of the country called a leach field). Seems like a lot of expense for a single toilet.
Next:  Your existing septic system might not be regulated (licensed) for an additional device. In my neck of the woods, septic systems are sized and permitted by the number of bedrooms in the house. I DK how a toilet in a barn would count.
There is also the possibility of a hybrid system, where you install septic tanks near the barn and pump it to the drainfield.
But and as MUCH AS I HATE TO SAY IT on a DIY site, septic design and maintenance is coming under increasing scrutiny everywhere. You could probably do this off the record, and not permitted, but it will catch up to you eventually. Better to do it right the first time.   You'll probably need to call a pro. Sorry!!!!!