Forgot to address this. In short, "balance" is not one class or race or build in a vacuum, it's the interaction of classes, races, feats, and spells, as a whole. The inclusion of something like Press The Attack and Fall Back shifts the balance and uniqueness of things like a Barbarian's Reckless Attack and the usage of reactions in turn for Fall Back. Spell changes alter the balance of caster classes. Feat changes alter the balance of martial classes. Adding in Level Up elements to an O5E game doesn't work. The rules all tie too closely together, they aren't designed in a "pick and choose" fashion. It isn't "one rule" therefore, but all the numerous changes to core rules and existing content that make the game incompatible with O5E in the sense that most players define "compatibility." A lot of OSR hacks are technically compatible with old school modules, but they are not "compatible" in the sense that they can be plugged into a game alongside original D&D player facing content.
A core 5e Fighter or Barbarian could see combat parity with a Level Up character assuming they have access to feats and mechanics as written in O5E, which complimented O5E classes. The same is true for O5E classes of all types. As noted, a game could theoretically have both side-by-side if the O5E builds are "only" O5E and access the original versions of Feats, Spells, Items, etc., and the Level Up versions are "only Level Up" content-based. Using the underlying Level Up mechanics damages the efficacy of O5E class functions, and skews the functionality of the game altogether. Trying to use them side by side as I'm describing is another logistics challenge, as monsters and opponents also have to determine if they are "original" 5e-based on Level Up based, but it would work better than applying the underlying Level Up rules replacements and still using O5E classes alongside the new ones. That would be the "worst case" way to use the new rules, as it would make the original builds worse, where the new classes are designed around the altered core rules. Such a usage doesn't actually reflect the true "balance" between the old content and the new, if the old content is worsened by new core rule changes, which it is.
Press The Attack and Fall Back are problematic mechanics entirely, however, and would need to simply be removed from the game.