Some additional comments:
I tried to find more information on your 7x12 lathe, and was not very successful. It may not accomodate the rear-mojnted, inverted parting tool that is recommended (and I agree that is an option if your lathe can do it. If you can do it, you may want to try it.)
To be absolutely clear, if you are parting, you should not support your workpiece from both ends, I.E.: with a tailstock center or with any other workholder on the end that you will be cutting off. I tried it when I was learning and what I learned was "Don't do that!" As you cut material away, the piece you are cutting off will want to fall off and the whole thing will break at the cutoff point and crash your machine, possibly taking lots of things with it and creating a chance for you to be injured.
If you are cutting a short piece off a long bar, you can support the bar in the chuck, and use a steady rest close to the cutoff point, but position it to the chuck side of the cutoff point. this will also have the advantage of allowing you to keep the end of the bar where you are cutting from wobbling. You may have an additional learning curve in adjusting the steady rest. If your bar is not round or is rough, You may have to turn the bar so it is round in an earlier operation, at least in the area where you will locate the steady rest. Many of the YouTube machinists cutting large round bars on large lathes often demonstrate how they go through this process.
I would recommend that you may want to try different cutoff tools, not just different inserts in your indexable tooling. It is true that you have to push carbide harder than HSS to keep a cut going and that may be hard with a small machine. If you have to use carbide, you may want to try different ones, as you have suggested you will do, but you may need some additional help in determining what to try, either by researching or by asking a good source. I have HSS, brazed carbide, and some indexable tooling for both lathe and mill, and I have found there is no perfect tool for all uses. I consider the cost of tooling to be sometimes an educational expense worth pursuing.
I do think HSS and a grinder are part of the learning curve, as you can really determine what works best in your situation with mostly only a time investment. (If THAT didn't work, Regrind the tool to a new profile and try again.)
With a small lathe and small projects, I found that I learned parting better if I could closely watch the cut: I have one of the circular lamps with a magnifier in the middle that I can position to watch the cut, and I also have a "Magnivisor" headworn binocular loupe. That way I can watch the chips being formed right at the edge of the cutting tool down inside the parting cut.
If you can't cut your piece off completely and cleanly, there is nothing wrong with using the parting tool to cut most of the way down, then stop and disconnect the lathe and cut the rest of the way with a hacksaw: Then all you have to do is face the cut to clean it up, which you might have had to do anyway.
I use lubricants more intensely with cutting off. I have settled on WD-40 for Aluminum and either WD-40 or "Tap Magic" for ferrous metal. Brass may call for experimentation because it is more "grabby" than other metals. This is on my small lathe. This is also with a traditional "T" HSS parting tool held in a holder for that type of tool.
--ShopShoe