Swingarm hydrometer is also pretty much useless for getting accurate sw readings. I would invest in a refractometer. Properly calibrated you will get accurate results needed for not only hypo (which requires absolute accuracy to work) as well as reliable readings for a tank that has inverts and corals.
I would still use refract and go with hypo. Your QT is not stable. You can place one on order thru internet, if you can't find one locally. In the meantime, use a glass hydrometer. They are decently accurate as well and will get you started with hypo. Hypo is the safest route in your situation. What is your current pH and salinity in the hospital?
How big are your fish? Obviously, smallest fish in the 10 gal. I'm actually going to say that you can add some LR in your hospital as long as you haven't started treatment. That will help out with establishing the biofilter and perhap keep that in there for a few days, THEN start treatment. With hypo you can leave the rock in (as long as it's just a few pieces of rubble rock, sm rock). You can also add about a cup of your sand from your display.
When you do water changes, and you will need to pretty frequently, use display water for the QTs (if water quality is good).