Sounds odd - I know that Luban is sometuimes a bit stubborn with SVGs, but Lightburn yet has to surprise me. That said, perhaps the same trick as with Luban works here also? Use Inkscape and convert all objects in the SVG to paths.
Regarding Constant power: As far as I understand it, the GRBL-M3 profile is basically only offering M3 - i.e. constant power. So that’s why there’s no switch - consider it as “constant power always on”-Profile.
I personally only recently played around with the different profiles, and with help from Lightburn support I found a working config - see here: Laser Frustrations and bugs - #4 by Aaron.F