...I'd like to see the group a little better
Sorry, but you're not going to like this answer; you won't see the holes any better no matter what scope you get.
I have a Vortex Razor HD Gen II, 3-18x50. It is a top-of-the-line scope. At 18x, on a clear and clam day with bright light, I can barely make out the holes from my .308Win at 300 yards. I certainly can't make any reliable decisions about the group with that scope. In fact, I've seen what I thought were groups only to realize that what I saw through the scope was not what was really on the target at 300 yards.
With a .223Rem (5.56x45) round you won't be able to see the holes at all at 300 yards. Sorry, this is just how it is.
A really good spotting scope might (and I emphasize might) work, but you'd need at least 60x. 60x is not uncommon with a good spotting scope, but you won't get one for $300 or less.
Just as an example, I've used this exact scope at a long distance competition:
It's a Swarovski ATS 65 (approx $2,500). At 65x I could see hits on a steel target at 450 yards, but couldn't really make out a group. The shooters were using either 6.5Creedmore or .308Win. A .223 in paper at long range simply can't be seen even with really good glass.
Now, you could go with an actual telescope. A reflector telescope like this might work: [ame="https://www.amazon.com/Orion-SpaceProbe-Equatorial-Reflector-Telescope/dp/B00D05BKOW/ref=pd_sbs_421_t_0/142-1883067-1238643?_encoding=UTF8&pd_rd_i=B00D05BKOW&pd_rd_r=77b9a868-e439-4710-a093-e30ad5eb894f&pd_rd_w=FfZL2&pd_rd_wg=D8MXU&pf_rd_p=5cfcfe89-300f-47d2-b1ad-a4e27203a02a&pf_rd_r=CWWDRY6SZCGH03MXZES7&psc=1&refRID=CWWDRY6SZCGH03MXZES7"]Amazon.com : Orion 09007 SpaceProbe 130ST Equatorial Reflector Telescope (Black) : Reflecting Telescopes : Camera & Photo[/ame]
The drawback to using a telescope is they're kinda delicate. So, you can't just toss it in the back of the pickup and head to the range. They must be handled with care. If you get one, let us know how it goes, I'm curious.
Part of the problem is price. At your budget, you just won't get decent glass. Good glass costs money.