Unfortunately you don't get the choice, the filament cathodes will heat up from the discharge and take preference over the 'cold cathodes' (anode strips) in electron emission, so you will still get the end blackening at startup. The tube that you're describing is actually an LCD backlight CCFL, which have cold cathodes and high brightness, with high frequency drive.
Mercury fluorecents do contain inert gas - Krypton for room temperature tubes and Argon for low temperature tubes, to initiate the discharge until the Mercury is fully vaporized (which happens very quickly). I believe SOX lamps include Neon in the mixture which accounts for the redish colour until they are warmed up - the Sodium takes much longer to vaporize as it needs a higher temperature.
Inductive compact fluorescent lamps do did / do actually exist, I think Philips made them. They were expensive though, and although there are no electrodes to fail, the phosphors still degrade in efficiency over time. They were a bit of a loss leader. The only other place that I can think of at the moment that uses inductive coupling is the Rubidium lamps in atomic clocks, where the ligh emission is narrow band from vapour itself rather than a phosphor.
P.S. I just looked at my 1927 Electrical Engineering book. Cold cathode mercury vapour lamps did actially exist back then (without phosphor), for industrial applications. The colour rendition was (predictably) very poor and the efficiency low.