Hey, Joseph!
Just two or three comments to add to the discussion. Before I start, let me say that most of the other climbers on this forum have much more experience than I do, so take what I say with a grain of salt.
First, what Sean said. If you've got a Rope Wrench or a Hitch Hiker, it enables you to choke the branch with a delta link or something similar, which can be loosened up and lowered down the branch as you progressively cut it back.
Second, this sounds like a pretty small limb to me, so I can understand feeling shy about depending on it as a primary tie-in point. Which brings me to the kick that Kevin Bingham has been on lately, which is the idea of climbing with two different ropes at the same time. Without seeing the tree, my thoughts would be that if I was going to attempt to do it, I'd have my primary tie-in point as high on the other branch as possible, with my first rope. Then I'd attempt to use a 2nd rope to do the SRT choke that was mentioned earlier, on the branch to be cut.
If you've got an SRT tool available, putting an ascender or a prussic hitch with a pulley above your SRT tool and running the tail of your rope through the pulley gives you a 3 to 1 mechanical advantage for pulling yourself out to your SRT choke point. It makes it amazingly easy to gain ground, even on a steep incline.
As long as your small branch doesn't break in the process of climbing out on it, your SRT choke climbing line will prevent you from taking a long swing back to the trunk. Just remember that if you do use an SRT choke, you have to attach some kind of a retrieval line to the side that you do not climb. In this way, you make a cut, drop down about 10 feet or so, or however much you're comfortable cutting at one time, tie in with a lanyard to secure yourself, then tug on the retrieval side of the SRT choke until it drops down to you. Choke it back up again and make sure your climbing hitch grabs, and you're good to go for the next cut. And so on, until you work your way back to the stem. I got this idea from Sean, AKA SouthSoundTree. I used it and it worked beautifully for me, but the branch I was working on had a much larger diameter, and was an oak, so I always felt really secure out there.
Hey, Joseph, I just had another idea, but it still involves having to get out on the limb at least halfway. There's this guy named Dan Holliday who has a website called ClimbingArborist.com or something similar. He's also posted videos to YouTube. Dan has a couple of videos that show two rigging techniques that might be useful to you in this situation.
The first is about Tip-Tying the branch, which would still require you to get about two-thirds of the way out onto the branch to girth hitch a sling around it. With a pulley high in the other branch, you could pull the whole branch up to the pulley, and then lower it straight down near the base of the tree.
The second method is to use a Spider Leg to cause the branch to swing out horizontally, missing the roof and then being lowered to the ground in an open area, still horizontal. This method is more likely to get hung up in branches if there are any in the way, but you might not have to climb out so far to make the Spider Leg work.
Here's the link to the Tip-Tying video.
https://www.youtube.com/watch?v=UbKZiejUuTg
Here's the link to the Spider Leg video.
https://www.youtube.com/watch?v=PtdabLZFwsE
Dan Holliday, (I hope I'm spelling his name right) is just an amazing guy, in my opinion. His videos were the thing that made me believe it was possible for me to learn how to do this stuff, as I had no climbers available to learn from. I think he recently joined the ArbtalkUK forum. I don't know if he's a member here or not; I sure wish he was.
Anyway, thanks for putting up with my long-winded post. Hopefully some small piece of it will be of some use.
Tim