This should be achievable with a single radio, if and only if you are willing to take time to really understand LoRaWAN, the other protocol, and whatever software stack you start from. It will not be a simple project, and you will probably not find guidance on the details, but rather have to figure out a lot on your own, starting with building a thorough understanding of how things already work.
As summarized elsewhere, your likely starting points are LMiC or LoRaMAC Node.
A traditional class A LoRaWAN node spends 99.5% of its time asleep, and only really operates in brief cycles between transmit and one or two precisely timed receive windows following shortly thereafter. All of the rest of the time, the radio hardware is yours to do with as you wish, within the limits of regulation and physics.
While LMiC is a complex codebase with some real oddities and hidden mechanisms, one thing it does having going for it is that it does most of what is needed to configure the radio each time it wants to use it. You’d have to make sure there is no other configuration somewhere but most of it is in the relevant LoRa and FSK tx and rx functions.
Another option would be to use LMiC (or whatever stack)'s existing transmit and receive functions for your custom purpose as well, though you’d need to figure out a way to make them take the air settings you are tracking instead of those the LoRaWAN stack is.
You’ll also have to consider the details of event scheduling - the problem is fairly simple - don’t let both things try to use the radio at once, but rather exploit the overwhelming idle time - but you will have to work out the details. Note that adding a second radio wouldn’t help much, as one shouting into the other’s ear would be deafening regardless of frequency.
It’s also unclear how far you have developed the non-LoRaWAN part of the system. That may be more complicated than it seems, not because LoRaWAN is special, but because it’s hard to coordinate a pair of node-class radios to communicate with each other while doing a polite job of using multiple frequencies. And this all but requires that your dual-mode device be mains, and not battery powered.
Realistically there are probably much better ways to solve your local communication problem.