Well, it depends on how much the noise bothers you and how much you like the sound the mic is giving you. You say it's not really noticeable once you start playing, so maybe just silence the part before the music kicks in, like I suggested.
As for a smaller diaphragm, they are pretty good for acoustic instruments because they have a flatter response curve and a wider pickup pattern. Unfortunately, generally speaking, the smaller the diaphragm, the higher the self-noise of the mic (though probably still less than your tube mic). Not that it will be all that noticeable, but that's just something inherent in capsule design, as I understand it.
If you do opt for a small diaphragm, there are plenty of options. I'm not familiar with the one you posted, but, depending on how much you want to spend, you can get some very usable mics. The MXL 604 and Studio Projects C4 are budget choices. The Shure SM-81 is a studio standard and one of my favorite
acoustic guitar mics. Audio Technica, Josephson, and many others make good mid-priced small condensers, too.