Investing in Proto Cu when you already have Cu

Not really. He is basically the same thing, substituting extra survivability with crit strength. Invest in him if you want, it's not like it'd be a waste - especially if you want to use a crit team, but he isn't necessary.

He fills a specific niche as a beast slayer servant.
Aside from his Protection from Arrows for his durability, he loses on debuff cleanse and Guts, which is what makes him a true undying Lancer breed.

His stats are pretty much the same as his original version but Prototype Cu specializes in crit damage as well. So it's not a waste to level him if you like him (even if you have Cu). I myself am attached to Diarmuid even though he's worse than both. I still like him and to me, that's all that i need.

Remember that "meta" servants are already strong in their positions even on their max levels so that Grailing them is usually not necessary (past level 90). Grail out of love, not gameplay. Play the game with characters that are fun for you, don't be too stressed over if you have the best meta units to "clear" game content.

Proto Cu is pretty similar to FSN Cu, with skills more skewed to crit ups and beast slaying in exchange for lower survivability (lacking Battle Continuation). So probably no need to develop Proto Cu separately unless you really need a beast slayer in your team.

Unless you want Cu chu train team go for it, but not really recommended. Like some people you can play it too and for example: i have Grailed OG Chu and max ascended Proto Chu after that borrowed Chu Caster from my friend, wreck some thing with this Cockroach trio( purely for fun purpose). Worse case when i enter troll mode, i just pour Nero, OG Chu and Heracles in a same team the result was guts everywhere. Remember just play for fun and don't be too stressed for it.

You should understand that Cu FSN's trait like amazing durability since you already have him in bond 9. However, his damaging is really mediocre, not matter poor performance of Quick card, or only 1 Art card hard to accumulate NP. And it is actually hard to put Cu in team because of his unique advantage from his skills is somehow double-edged swords. This could be a critical because some challenge quest in the future, enemies will posses amazing HP with multi-hp bar and become more annoying like sure-hit or remove buff of Cu's.

Then Cu pro become better choice to deal decent damage. His 1-3 skills can make combo to any Beasts, Berserkers and Archers. Having skill with combo of star-adsorption and critical-up are all DPS servant's dream, let alone Cu Pro's 1-3 skills combo has 3 turn long-lasting, 5 turn short cool-down, and in same pace.

However, the basic attacking status of 3 stars is still not comparable than SR or SSR, even though you may grailed Cu FSN or Cu Pro to lv90, still not high.

So if you dont have too much SSR or SR Lancers/Berserkers, or posses Lancers/Berserkers with quite different role like Liz or Nightingale, then leveling Cu pro is also a good choice. Right now it is not recommended to have Cu pro to above Lv60 due to his S curve growth, and skill-up besides of his 2nd skill. After his strenghening in the near future, you can eventually upgrade his 1-3 skills.