Going to take the Milcun advances marksmen course
OK so the question you are really asking is, "should I shoot (several hundred rounds of) .223 or .308 during my Milcun course?". I suggest you ask Milcun for their recommendation. They are a first-rate outfit and I would trust their guidance.
To shoot many hundreds of rounds of good .223 ammo vs. good .308 ammo involves many hundreds of dollars of expense, so this is a reasonable question. Keep in mind though that your ammo expenditure is only a (small!) part of your total cost, you will be spending several days of your time, and your course fees will likely be several times more than the ammo you burn. It won't be a cheap experience, but it will be a very high quality one. Choose whichever ammo is most appropriate for the job, even if it is 1.5X more expensive than something else.
(and BTW there will be some cases in which .223 will be preferred over .308, and vice versa - it's not a question of the more expensive .308W ammo being "better" in all cases).
A great deal of .223/55FMJBT ammo out there is "crap" from a target shooting point of view... that doesn't make it bad or unsuitable ammo, for many purposes ammo that feeds and goes "bang" pretty much every time serves the purpose well, even if it shoots 5" or 6" groups at 100. This also doesn't mean that a .223 is inaccurate, or that 55 grain bullets are inaccurate, or than FMJBT bullets are inaccurate. For example I have shot Nosler 55 grain Ballistic Tip bullets (a plastic tipped bullet with a mild boattail) at 600 yards from a bolt action .223 with a 1-9" twist barrel, and they shot very well (as in, up to target-grade levels of accuracy). They got blown around by the wind more than you would get with a more typical long range match bullet that you would typically use (an HPBT bullet of 68, 69, 75, 77 grain weight).