Trenching up against fence posts
You haven't done yourself any favors, but if only the posts are in you could make it work. You'd just have to plunge next to a post, trench six feet, then rotate the machine and repeat. The blade ought to protrude enough to clear a six inch post, and if the fence sections aren't installed clearance between the posts isn't an issue. 

In all honesty though, I suspect using a mattock to dig pass the posts may be less aggravation. Trench what you can, and knock the gaps down by hand. It's not like you need to get all of the dirt out as you can work the mesh into loose soils.

I hesitate to mention it but if it's just the posts and they're set in the ground not cement you can pull them straight out and drop them back into the hole after you do the trench. The easiest way is definitely a front end loader with a chain on the bucket; it's a lot more work with a chain, crowbar, and blocks.

There are also chainsaw attachment/chainsaw type trenchers. (Terrasaw among others) I've no idea if you can find one at a nearby rental place but it may be worth a look. I've no experience with them, sounds like a long day but it'd get you close to the fence. [There's also something called the "Alpine Magnum" that I mention for the sake of completeness. I'd never even heard of them before and I'd be absolutely amazed if anyone rented them.]

If the whole fence is in or nothing above works I'd suggest renting a small excavator for the day, assuming you have room along the fence line to drive. That would let you dig flush with the fence, but the trench would end up a bit wider.

Maybe the best piece of advice is to visit your local equipment rental place (the one the specializes in renting machinery, not the BORG) and explain your problem rather than what you want to rent. They know what they have in inventory, and can give you suggestions. For example, the one here rents Bobcats for 1.28 Gigadollars per day and small banged up Kubota with a backhoe for about a quarter of the cost. So "I'm digging a hole" saves you a ton of cash over "I'm looking for a bobcat excavator." Maybe you'll get lucky and they'll have an offset trencher or small skid steer with a tiny bucket.