Should I raise Best Doggo ?

Yes - he's great.

Why wouldn't you level someone you tagged as "best"? Just do it.

you have many alternative, so i would say he is not an urgent servant to level up. but eventually why not?

Cu's closest competitor in the servants you have is Enkidu. Post strengthening (or interlude, I can't remember) Enkidu gains a single turn evade on a 5 turn cooldown. That coupled with his 10k heal gives him survivability comparable to Cu's, but Cu's PfA is just a better evasion skill, and while disengage C isn't even close to the magnitude of Enkidu's heal, you'll actually be able to use it more than once a fight, unlike the 10 turns on Enkidu's skill. Cu's NP also does comparable damage to Enkidu's at level 100, and it's got the benefit of a sure-hit buff post strengthening, while Enkidu has a near-guaranteed stun vs divine enemies. Cu is lacking in a damage steroid, but in place of that he has a guts on a 7 turn cooldown.

Essentially, if you're facing something divine, use Enkidu, but if you're vs an archer without the divine trait, or anything but a saber, then use Cu. His damage is good enough for a main DPS, and his survivability is insane, making him a good choice for an anchor or soloer.