What you need is a curved needle: it will allow you to sew from the top without ever needing to have access to the underside. They are cheap and widely available. When you use one to sew your patch to the pocket, you can use your hand to keep the pocket expanded, or even better put something small and solid in it to keep the two layers separated while you whipstitch the patch down from the top. Watch a video on how to use curved needles for repair.

Going by name, there are two kinds of curved needles sold: upholstery and embroidery. You absolutely want the embroidery kind: the upholstery needles are huge, made for much larger projects, and would make the fine work you are trying to do very difficult. Also, if it's there (doesn't look like it is on your patch) you can pare down any hard coating on the underside of the patches around the edges without disturbing the patch itself, which will make it easier to get a needle through the patch when you go to put it on.

And wax your thread: pull it lightly across a candle to coat it a tiny bit, then pass it between two fingers to soften it into the thread. (There are also commercial products that you do not have, but a candle is fine in a pinch.) This keeps your thread from tangling more than it has to, and lubes it a bit for the sewing. It's not essential, but it makes everything so much easier that I never hand sew without it.

If you get a thread that closely matches the beige of the patch it won't matter how sloppy your stitches are, because they won't be seen. But if you can, wash it and clean the grime off as much as possible before you do this, because it's hard to match thread to dirty fabric. Also, when you are trying to match thread, sometimes value (light and dark) matters more than color. Because you're sewing a lighter colored patch, if you can't find an exact match, choose the closest one but go a shade lighter: the dark will stand out more on a lighter background. (It's the opposite for darker materials, where you err toward the darker thread instead.)

If you have any problems or questions, feel free to hit me up. I've been doing my own hand and machine sewing for decades.

EDITED to add: Here's a brief video on whipstitching a felt patch: https://www.youtube.com/watch?v=3NLr1r5ZMAM - she's not using a curved needle, but this is the stitch you want.

A video showing how to used a curved needle for repair is here: https://www.youtube.com/watch?v=WJUuPHDPTyc. He's using an upholstery needle with pliers, you can see just how big and unwieldy that thing is, but that's essentially how you use a curved needle to sew from the top only on pretty much any project.

Honestly, I have no clue what thimbles are for. But I realized at some point this year that instead of using some hard surface to push my needle through assy material and/or hurt my finger pressing too much, I can just... Use a metal thing that sits on the right spot.

Also I know this sounds weird but go to a shop and try them on. It's weird how much a good fit helps.

The cutting out would probably be the way to go. I mean you could flap it, so it stays connected to the right spot, maybe it's time for you to be brave and risk it. Especially since you have a sewing machine! That means the seam can turn out quite nice when put back in place. Anyway, good luck and updates are appreciated!