IF you were doing your CQB at a reasonable pace, you would be going at a very slow speed as compared to what you would be running in 3Gun. Even the 50% shooters in 3Gun will be engaging say 5 targets (10 rounds at say 5-15 yards) in about 5 seconds, the top guys will run that in about 2.5 seconds. Two optics (red dot and scope) put you in open where the cost and speed is even higher. Side irons are slow as compared to an optic with a true 1x at the bottom, but faster than trying to use a 2, 2.5 or 3 at the bottom end of the magnification. Most people tend to over-magnify. I used a 3-9 at my first few matches, then 2-7, then 1.5-6 and the last 4 years I have run 1-4s. My scores improved every time as I got closer to the 1-4. I have placed in the top 10 on long range stages since going to 1-4s, and that includes targets out to 600 yards.
All shooting is always about fundamentals, so don't fall into the spray and pray trap. The 1x allows you better vision with more speed than a magnified optic. You still have to have a good sight picture to press the trigger and get the hits. The 1x just lets you see faster.
The "budget" is probably what is getting you. The Burris MTAC is very good, if not the best, budget optic for 3Gun. The Vortex is very good as well, but the thinner reticle, for me, is slower inside 100 yards which is where most of the shooting happens in 3Gun. The Vortex thinner reticle might be a tad better at 300 yards and out, but I get my hits with the MTAC (unless I blow off the fundamentals, and then no scope helps).
When I shoot customers guns to check groups, I put a fixed 10x optic on, and my 100 yard groups go from about 1.2 to 1.5 MOA down to .5 to .8 MOA, so magnification does make a difference. I just don't think the balance between a precision optic and a 3Gun optic comes together until you cross over the $1K threshold, maybe even higher.
That said, for about $800 or so, you should be able to put a 1-4 on your AR and have a 10x for precision. With good mounts, they are repeatable.