I don't have my (TSO'd) SkyBeacon unit yet, though I'll bet there's people on this forum that have the Experimental model. But I do know you need to have the nav lights on to have the SkyBeacon working. So, turning them off would disable it. I'll let someone else address the wisdom/legality of flying (even taxiing) with your nav lights off.
But I believe it's all moot. I know from their doc that there is a configuration option in the app that controls the SkyBeacon that allows you to put the unit in Anonymous Mode. Whereupon it will not transmit your actual tail number or your plane's unique ID code. What I don't know, and would like to hear from someone that actually has a SkyBeacon installed: does that mode stay enabled even after you've cyled the SkyBeacon (turned it off, and then later turned it back on)?
It is far better for you to have your nav lights on, and to have your altimeter reporting, AND to have ADS-B Out working. You want to be seen, don't you? Also, without ADS-B Out working, you're not getting traffic (TIS-B) sent to you -- you're only seeing pieces of other planes' (that have ADS-B Out) traffic around them. (There's another question: if you're in Anonymous Mode do you get TIS-B transmitted to you?)
BTW: I believe that Anon Mode is only available to units that use the UAT frequency (like SkyBeacon), as opposed to Mode S units (such as Stratus ESG) which cannot have that mode. On the other hand, UAT units do not meet the requirements of other countries that have implemented ADS-B, which all require Mode S. Such as Mexico, and as of this stage of their planning, Canada. (Though I hear Canada may make an exception for around their southern border to allow for UAT ADS-B).