I think the OP is suggesting that the volume of water in 45ft of trace heated cold water pipe is enough for a quick shower when combined with the hot water feed to the shower. Trace heating seems an expensive way of doing it though as the large surface area will have considerable heat loss even with insulation over the tape.
I have a similar problem with some hot water taps being 68 ft of pipe away from the hot water storage tank. I put in a circulating loop which works well but the heat loss is horrendous. The only economic solution is to generate hot water closer to the point where it's needed.
I believe your last comment is the only option in the OP's situation.
Unless I missed it (highly possible

) I don't think OP has provided us enough information. I think the only one who thought about trying
heat trace (not what it is made for) was the OP. But, I get you. Closer is always good.
A larger
(proper sized for application) heater / tank , set hot enough would be good too. Sounds like that may not be an option for OP since a landlord is calling the shots and may not want to be bothered with it?
One time I had to install a pump and design a continuous circulating loop , similar I think to what you mentioned at one of my job sites.
The pump and a circ loop provided instant hot water to all access spots
(showers and wash basins). We also upgraded to just a little bit larger
(not a really big nor expensive) water heater tank and made sure mix valves were set above minimum setting when installed.
Once the
combination of things were in place , problem solved. Another big thing that helped us was that all of that system + pipe , was inside a very well insulated locker room.
Unfortunately for the OP , it sounds like he does not have many options at all due to someone else owns the place and seems to want to charge the OP for any repairs or upgrades. When I had to rent , years ago. I had an agreement with the owner. Anything needing repairs , if I showed him first and did the agreed repairs myself , he took the cost off my monthly rent. I realize this is not an option for everyone.