In fact, everything is much simpler.
There is a script /etc/init.d/S99custom in the system.
If you look into it, we will see the following (only important part is shown here):
start() {
for i in /etc/kvmd/user/scripts/S??* ;do
That is, it searches for user scripts in this folder - /etc/kvmd/user/scripts/ (needs to be created manually), which must match the following mask S??*.
So everything is simple, we make a script, call it S20tailscale-update.sh (for example), put it here - /etc/kvmd/user/scripts/, and everything works and survives updates.
This is how I managed to add support for the Wi-Fi USB adapter that I posted about here on this forum.
And the most important thing in your case is that this script is launched just before Tailscale is initialized.