Sponge Filters are so cheap. Why DIY?
Besides if tank is fill and filtered or just with an airstone, it will cycle without doing anything to tank.
It may not have amount of BB as if one cycled with fish/Ammonia, but it will be bio active.
If you are to use 55 as Q/T, Why not filled it up and age the water as reserve for pwc until need to utilize 55G as Q/T.
When need to use reserve as Q/T or as Hospital, swap bucket of water at a time b/n main & reserve tank. After good work out(lol!), water b/n main and reserve should be very similar. then move the fish
Once Q/T is well served and all habitant transferred to main, you can heat up the Q/T water for few days beofore dumping. To sterlize the 55G, I would just use salt water to sterilize rather than bleach (unnecessary pollutants to environment). You can even use rainwater + salt to sterilize 55G. Free!!! If have room in backyard in the shade, set up vat with lid and diy contraption to fill and drain (control valves, etc).
Want to grow best live fish food, trays with couple of dried leaves will be perfect breeding ground for insect larvae.
I am surprise that not too many people use rainwater, especially in the regions with high pH/very hard tap water. Its free and probably better than tap! EXCEPT if one lives in/near industrial area then I would not trust rain.
Anyway, Sponge filter at lfs should not cost more than $10.00, may be couple of dollars more. Tetra Brilliant SPonge filter (I think this was the name) was about $8.00 about 7-8 yrs ago. Replacement about $5.00. Depending how wepp maintained, sponge will last long time. You will probably need strong air pump to drive SF in 55G.
Good Luck!
btw, Only draw back of 55 as Hospital Tank, cost of med will be way too high. You can filled 15G worth or 20G worth, etc. etc., depending on amt of fish to be treated.