I'm going to work out a plan for you that involves your taking on the least expensive actions possible that also help to solve your issues at hand.
I'm not sure why it's not plainly obvious to others that your completely tied to a budget solution. Your first post pretty much spelled it out....with your obvious joy at having acquired such an inexpensive Screen.
.....and the following post with the image of the room / screen? I immediately guessed you were in a Rental / temporary location, because you had stated that you "could not" mount the PJ permanently.
1 + 1 = 2 ....., and both those posts of yours add up to your need to find a real, inexpensive solution to a problem that prevents even your NOOB's eyes from enjoying your purchases.
......and no calibration is going to solve the all too obvious issue of wash-out. Nor will simply reducing the PJ's output be a satisfactory answer. While that would actually help a little bit, I'm sure the loss of brightness would itself become a sore point...so that is not the answer you seek.
Under such circumstances, it would seem fruitless to suggest that someone spend a couple hundred dollars plus for a Calibration Tool/Software....or even $40.00 for a even simpler dedicated Calibration Disc, just to tell them something that is so patently obvious.
It's the Room that is at issue....not so much the PJ/Screen combo. And the Room cannot be changed. (...well maybe it can...but that is to come later...
So OK...I've been there and done that so many times before it's silly. In 10+ years of over viewing Member's situations, I tend to cut to the chase. While that can trouble some, and even irritate others, the express purpose of my doing so is to get people started and on their way to a effective solution. Makes no never mind as to what paint, material, or room alterations are suggested...it's all about making the necessary changes possible. Otherwise...nothing get's accomplished.
For the record, both the other Posters are absolutely correct....If you don't do anything else, you need to at least get a animated Pixar DVD and use the THX Optimizer program included. Or...if and when you coat the screen with whatever you determine to go with, you'll need to calibrate it to the screen's characteristics and to assure that the PJ itself is color corrected.
First off.....that damnable White Room. A Rental, eh? Well....look around at the nearest Fabric Store and try to source out some Black Felt. ( 8 yds should do it...) Get some Thumb Tacks. Stretch out and apply a single width's strip of the Felt across the Ceiling over the Screen. Do likewise with the Walls to each side. Don't bother worrying with the Screen wall...it has no bearing or effect on the Screen's output.
Just that simple solution in and of itself will immediately change things for the better. The Thumb Tacks won't mar the walls or ceiling....so don't let that stop you.
Now...to further your end results....remove the Screen from the Frame. Drape it against the Wall, securing it first with a minimum amount of Thumb Tacks, then sealing the edges with Painter's Blue Tape. Go get some Dulux Interior "Light & Space" Ultimate White in Matte, and have it tinted to a N8 Neutral Gray.( ask for "Universal Gray" ) Get at least two, "GOOD QUALITY" Rollers of a Low Nap for smooth Finishes. Make sure the Roller Wand is a good one as well...one that spins very easily. Get a Large Sanding Sponge (Fine Grit...look in your Home Supply store's Drywall Finishing section...)
Thin the paint with just enough water mixed in to allow it to pour...not "slide' out of the can. The Dulux paint is almost sludge-like...it needs to be thinned.
Take the first Roller Cover and before you put it on the Roller wand, dampen it, express out the extra water from the Roller, then push it onto the Roller wand...load it up with the paint, and roll out a few strokes onto a sheet of something (...buy a piece of Drywall...) to get the paint evenly distributed throughout the Roller. Doing this will allow you to judge how the paint is going on at the thickness you have it. This is important because even with the material tacked and taped to the wall, applying too much pressure to try to roll out paint ridges can cause the material to buckle in front of the Roller. That can be / is a PITA.
Now if you feel confident in how easily the paint is rolling out with light pressure, roll on a light coat. Let it dry "COMPLETELY" overnight. Roll another coat on. Let it dry. Now very lightly sand it using smooth, long sweeping strokes. Wipe off the dust. Break out the second new Roller cover. Wet it...prime the roller, and do your best work on what will be the last coat.
I've said it before and I'll say it again...while many advocate rolling, I personally have found it fraught will potential issues. And buying the "right" rolling gear isn't really all that inexpensive itself. But overall, it's going to be less expensive than you shipping in the "No Name" Sprayer.
But....if you want a virtually perfect surface, and avoid the potential Slings & Arrows with rolling paint, the extra $120.00 for the Gun w/Shipping will pay real dividends, and I'll betcha you'll find other uses for the sprayer...or in the least find another willing buyer for it at $50.00.
There are a lot of your Mates in Australia who are AVS members, and it just might be that someone would jump on the chance to get the No Name off you, like a Funnel Web Spider will jump on a bare Toe.
'Cept you'll be happy they did...and you won't wind up dead.