Since you’re taking the trouble to set up a monitor, I should be nicer!
The Video / Broadcast Colors filter, like a limiter on an audio circuit, can do a useful job in clipping the signal to keep it “legal”; but it can cost quite a lot in picture quality, particularly when the signal is quite a bit out of legal spec, which I guess is your situation here.
You may not need extensive grading in Color Finesse, though, if your material looks good in RGB / computer colours. The Effects / Adjust / Levels filter will let you set input and output levels: you could set this to output black at 16 or so and white at 235 or so and see whether it does most of the job for you.
Some video edit software allows you to choose on import whether to treat graphics as having full RGB colour values or the reduced video-friendly set, but I’ve never found this option in AE.
On the encoder side,though, the CinemaCraft Encoder Basic version is cheap, fast, and produces very good quality output (so long as you don’t need Dolby AC3 or can do that in something else – for us in PAL land, it produces good mpeg audio.) And in its advanced settings panel it offers a choice of interpreting incoming signal as “16 to 235” or “0 to 255” – which just might be an alternative for you …
All best