Is it normal for my battery voltage to sag that much when the rig keys up?

Some sag on voice peaks is completely normal, especally with a small battery. SSB doesn't draw a steady load like FM or digital modes do. Your average draw on a normal SSB QSO is way below peak, but every time you talk into the mic the rig pulls a big burst of current for the duration of your syllables. A small 10ish Ah battery, particularly if it's a sealed lead acid type, will dip under that kind of momentary load even when it's fully charged and healthy. If it bounces right back the second you unkey, that's the battery and the wiring doing thier job, not a fault. That said, thin wire can absolutely make it worse. Voltage drop in a DC run goes up fast as gauge gets smaller, and it compounds on both the positive and negative side. On a short run in a box it may not be dramatic, but every bit of it adds to the dip you're seeing on the meter. The HF rig is the one that matters here. If some of your accessory leads are thin, that's fine, but the main feed from the battery to the HF rig should be the fattest pair in the whole box. I'd also check every crimp and powerpole connection while you're at it. A loose or poorly crimped connector acts like a resistor and causes sag that looks exactly like what you're describing.

Easy way to isolate it: put a meter directly on the battery terminals and key up. If the battery itself barely moves but the panel meter drops a lot, your wiring and connections are eating the difference. If the battery sags hard too, then it's mostly just battery size under peak load, and the fix is either a bigger battery or accepting it, since the rig clearly doesn't care. Also worth checking sag with the battery freshly charged versus partway down, because small lead acid packs sag much harder as they discharge. If it's only ugly at lower charge, that's your early warning sign in the field.

Mine did the same thing and it turned out half the sag was the meter itself being wired way down the line from the battery. It was reading after a fuse block and a couple of connectors, so it saw every bit of drop in the run plus the battery's own dip. Moved it closer to the battery and the number looked way less scary. Worth checking where yours is tapped in before you go swapping wire. Also, I ended up tossing a LiFePO4 in mine and the sag basically vanished compared to the SLA I started with.

I'd push back on the 'the rig clearly doesn't care' part. Mine cared plenty. On a similar portable setup the sag got worse as the battery ran down over an afternoon, and one long over transmission was enough to trip the low voltage warning and the radio rebooted right in the middle of a QSO. That's a lousy thing to find out at a park an hour from home. So even if it seems harmless on the bench with a full charge, I'd fix the wiring anyway and treat the worst-case sag at half discharge as the real number to design around, not the fresh-off-the-charger reading.