What is the difference between "xxといかせてもらう" and "xxといく"?
This title may attract duplicate flags, because there are posts out there that answer the question literally. Your actual question is quite specific and a "good" one tho.
So can we use 持久戦といく in this context?
Yes, we can.
If we can, what would the difference between 持久戦といかせてもらう and 持久戦といく in terms of meaning and nuance?
Or more generally, the difference between "xxといかせてもらう" and "xxといく"?
I'll use your example. Despite the use of 戦, we know it's not about a battle, but instead it's about a strategy to consider the long term. But for the sake of explanation, let's treat is as a battle.
In a battle, a commander may be straightforward and say "I shall initiate a long term strategy". But I bet you've also heard phrases like "well well well, why don't we shower them with a gift of endurance and pain" or "allow me to introduce them the taste of long lasting suffering".
They mean the same thing in the end, but the nuance are different, don't you think? When one declares a tactic by saying いかせてもらう, it may be expressing sarcasm, crookedness, dramatic effect, etc. It might as well carry the effect of sounding smarter, clever, or cunning.
It depends on the situation. Take Freeza, a ruthless feared villain from Dragon Ball Z as an example. He (basically) adds layers upon layers of words to sound kind, calm, and polite - させて + いただき + ましょう + か - which ironically amplifies fear to the readers.