Is it normal for satin to pucker when I hand-stitch a side seam?

The puckering is almost certainly the backstitch, not your tension. Hand tension isn't really a thing the way it is on a machine, since you control every stitch by feel. What's happening is that backstitch locks each stitch down tight with no give, and satin is a fabric that shows every bit of stress. Even if you're not pulling hard, the thread is sitting at full tension on every stitch, and that's what causes the ripples. Small stitches actually make it worse because you're cramming more tension points into the same space. If it were me I'd pick the sides out and redo them. Two options: a running stitch with a longer thread and very light tension, which leaves the seam a bit flexible, or small pick stitches, which look like machine stitching from the outside but only catch a couple threads of the satin at a time so there's nothing to pucker. Either way, baste the seam first with long loose stitches while the dress is hanging or on a dress form, so you're not fighting the fabric while you sew. A finer needle helps too, something thin and sharp, because a regular needle is shoving the satin threads aside and leaving little dents along the line. On the lace, you've got the right instinct about not stitching through it, but don't just leave it floating. Once the satin and lining seams are done and pressed open, the lace panel will have extra width where the seam used to be wider. What you do is lay the lace back down over the new seam, smooth it so the pattern flows naturally, and tack it down with a tiny ladder stitch or loose whip stitches that only catch the selvedge edge of the lace. Done that way the tacks basically disappear into the lace texture and nothing sags. Pressing matters more than people expect with satin. Use a press cloth on low heat and press the finished seams open flat before you judge how they look. A seam that seems lumpy while sewing often settles right down under the iron. With a few weeks of runway you've got plenty of time to rip and redo, and honestly the second attempt will go twice as fast since you'll know exactly where the seam sits.

I did the same thing on my sister's bridesmaid dress last year, took in the sides by hand because I was too cheap to drive to a tailor. What saved me was testing on a scrap first, I cut a strip from the hem allowance inside and practiced the stitch there til I found a tension that didn't ripple. Way better than discovering the problem on the actual seam. Also second the pressing advice, I used an old pillowcase as my press cloth and it worked fine. One thing I'd add is to try the dress on after basting but before the final stitching, because what looked even on the hanger pulled weird once it was acutally on a body.

So glad this got updated with the wedding actually happening, congrats! I'm in a sort of similar spot taking in a dress for a friend, and the part about smoothing the lacy top layer back down and tacking it confused me a bit. When you say the stitches disappear into the texture, do you mean you literally sew through the holes in the lace pattern, or just kind of catch the edges where it's solid? I tried catching the edges on mine and it kept slipping off the needle. Also for the basting first, is a long loose stitch just a regular running stitch but barely pulled, or is there a trick to it?

Agreeing with the backstitch diagnosis here. I made the same mistake on a satin cocktail dress years back and ripping it out felt like a huge deal but the redo took half the time like that first reply said. I went with a loose running stitch even though I was worried it wouldn't be strong enough, and it held thru a whole night of dancing with zero issues. One thing I did differently was anchor the start and end of each side by knotting into the existing seam allowance instead of the satin itself, so the knots never showed on the outside.