How can I switch a circulation pump between continuous power and thermostat control?
Yes you can do what you want.  If you don't have zone valves and you don't at least temporarily have the ability to move or insulate the pipes or provide a better long term solution, you can do what you suggest to keep water circulating.  You don't need to switch the neutral.  Just put an ordinary light switch in a properly housed and wired box to connect L1 to the hot side of the circulator, probably C1.  For wifi control use a zwave switch or any similar device, there are all sorts.  Move the circulator cable to the new box so you can use its knockout to connect the two boxes.
There is nothing on the low voltage side of this controller that would allow you to switch on the circulator without also turning on the heat.  The manual switch just bypasses the thermostat to turn everything on.
Note that with your situation you will actually be chilling the house, circulating near freezing water from outside to all the radiators, and hopefully eventually triggering the thermostat to fire up the boiler rather than just freezing the entire system.  Good luck with that, you might be better off disconnecting the offending section of pipe and putting an electric space heater in that room until you can address this properly.
Another thought, if there is a spare wire in the thermostat cable you could do this with a fan relay instead of a light switch and use the thermostat's fan control to control it.  Most stats have a &quot;fan always on&quot; switch and some have a mode that will cycle the fan randomly when heat is not called for.