Just to report back, it worked perfectly. It even now picks up the lastest value properly, whereas I'm pretty sure it didn't before. Very nice!
So here's my take on how to do switchable LFO's:
You primarily rely on the built in modules monophonic/polyphonic flag for reset/retrigger behaviour. Don't bother even building it for your own network.
Do your sync this way:
0bc0eeef-5d7a-4c85-822a-a9b901fc0519-image.png
A branch container, containing two chains. Each one has its own ramp source in it. Make one of them the regular ramp source, the other one the clock_ramp source.
You can even write a simple math expression to adjust the phase.