That is entirely dependent on him not believing his son an absolute screwup- which if he did he would very likely not have grafted that personality override in the first place. This one is up in the air until we know for sure (which will probably only take a few pages, really).
Klaus
knows his son is not a screwup. He set up elaborate tests for Gil's development using Dr Beetle and others. Later he proved his competence very spectacularly at the gates of Mechanicsburg. Klaus was rightly proud of that.
Gil has fought Othar enough times to know him, Klaus allowed Othar's escapes to test Gil and was confident his son would not team up with him. The only concern Klaus has is regarding Gilgamesh's emotional entanglements. He is in love with a dangerous woman channeling The Other, and smitten enough that knowing that doesn't give him pause.
Klaus trusts his son almost completely, standing orders were given for transfer of control of Wulfenbach assets to Gilgamesh, which was enacted on Klaus's self imprisonment. There was a rumour, probably cultivated by Klaus that his heir was a lunatic who had to be kept holed up in castle Wulfenbach. Pinkie believed that one so likely many of her employers did also. We know also that Klaus smirks when people understimate his son, and warned Bang not to do so.
I am convinced that the personality override has nothing to do with Gilgameshs competence and fitnesx to rule but is there to prevent Gilgamesh from being infleunced by the Other. As final proof he was willing to release control back to Gilgamesh to fully employ his abilities as a spark.
Now I am biased here, Klaus and Gilgamesh are my favourite characters. If I somehow tumbled into this horribly dangerous world I would sign up to team Wulfenbach over any other choices.