Okay, the Rolex is a certified chronometer, and the Speedmaster is not, so those are apples and oranges to begin with. I don't know how many "a few" is, so I can't really relate to what that is in a month. However if "a few" means 3 minutes, then you are certainly pushing what this watch would likely run at out of the box, because 3 minutes a month is essentially within COSC specs. If "a few" means 6 or 8 minutes a month, then I expect this would meet that requirement.
Seriously I'm trying to help you, but you are making it difficult to give you solid answers...
Anyway, the link provided tells me it's an Elabore grade movement, so it does not use the better balance wheel, balance spring, pallet, and mainspring that the Top and Chronometer grades use. The 4 readily made grades are Standard, Elabore, Top and Chronometer, so this is the second of 4 grades. However as it happens I have serviced one of these models (not the all white version, but the same model with blue/black on the dial and bezel) and I was able to get very good timing results on the watch. As you can see the movement has zero decoration:
Disassembled:
Timing in 6 positions:
Results are excellent:
To put these results into perspective, the Elabore grade movement allows the positional rate to vary as much as 20 seconds measured over just 3 positions. Here I have only 6.4 seconds over 6 positions. For an Omega COSC watch, they allow as much as 12 seconds measured over 5 positions, so this is an excellent result. Note that this is more than just regulation to get the watch this good - it requires being able to adjust the watch, which is a completely different thing to regulation. With just regulation as you noted, it won't be this good.
Pressure tested and done:
Some brands pay a lot of attention to timing on their watches, but not most of the smaller micro-brands like Steinhart. You might get lucky and have one meet your requirements right out of the box, but it's a crap shoot. Some brands are worse than others - Oris I swear does zero adjusting on their watches and just installs whatever they get from the factory...
So if it's handled by someone who knows how to properly adjust a watch, then it could be as good as a COSC watch even with the lesser parts.
Hope this helps.
Cheers, Al