@Michael Martinez yeah that tracks — the supercap's initial charge inrush is the culprit there. During that first few minutes it's essentially a near-short to the charging circuit, and if your USB host is also trying to enumerate the STLink at the same time, you're competing for current headroom on the 5V rail.
Easiest workaround during dev: let the cap charge up first before you plug in for flashing. Once it's topped off the draw drops way down and the USB connection should be rock solid. Five minutes is all it takes to get to full anyway, so it's not a big workflow hit.
If you need to flash repeatedly in a tight loop and don't want to wait, powering the cap from a separate 5V source while using USB exclusively for programming would fully isolate the two — keeps the STLink happy and the serial monitor clean.