The decision is yours due to two factors:
1. I don't know how much ambient light will be in the room.
2. Personal preference.
If I had a 5050UB I would want to get the best blacks, which would exclude an ALR screen. Even though movie watching is only 35%, PQ in the dark would make it worth it. With some lights on it would probably be fine. Depends what content is being watched. Bright content will fare better than images with dark parts, which will be the most affected.
I recommend you get samples, or go for the paint option.
Additionally, to get the best blacks this projector can provide I would use black velvet around the area of the screen, 5' or so, including ceiling, and a black carpet on the floor.
If you're continuing chasing blacks, a B stock JVC NX5 can be had for around the price of the 5050UB. It will not be as bright as the 5050 and 130" will be the limit, but with a treated room and only some lights in the viewing area it should be fine. @skylarlove1999
might disagree. The JVC also has a better HDR tone mapper.
Personally I would make this sacrifice for the superior PQ, but it's YOUR setup.
A 1.3 gain white screen or the paint option would make the JVC setup a bit brighter.
The ALR's mentioned in the first post would have the most ambient light rejecting properties, but while they may be better than a white screen, they would be overkill for that room.
Besides the paint, an intermediary solution would be High Contrast Grey from Silver Ticket. It's got 0.67 gain, but it's not just a grey screen, it has ALR properties (negative gain ALR screens exist).
This fabric will improve the black level and help fight ambient light. The JVC at 135" might be too dim, but the 5050UB can do it. The probability of hotspot will be very low.
I've mentioned that it's a decision that you have to take, but since you haven't seen what a projector with good blacks looks like it may be difficult.
I also assume this is the first projector setup.
Another recommendation would be to get a budget projector and use it on a wall for about a year or so, to get familiar with projection basics. It may be a safer alternative then jumping into ALR's and high end Epsons/JVCs.
The Epson HC3100/3700 would be great units, for cheaper but with worse black levels the HC1040 and HC1060.