Gruumsh has a particular hatred of Corellon, god of the elves.
I agree with your description of Gruumsh. I don't have Volo's Guide, but there was a similar description in "Deities and Demigods". I would run Gruumsh as described. Gruumsh hates elves even more than Churchill hated Stalin.
clerics strive to embody the handiwork of their deities.
True, but PHB also says that specific overrides general.
If the OP DM has ruled that Gruumsh accepts the worship and service of this half-orc who's rejecting human culture, then the OP DM has made a specific ruling. The DM has said that as the cleric reaches 3rd level, the connection with Gruumsh deepens; that connection is *only now* reaching the level of conflict you describe.
There's a huge difference between the distrust between Legolas and Gimli and someone joining your group who says "Oh, by the way I'm the acolyte of a murderous sociopath. I'm a firm believer in killing you, your loved ones and your children when we get the chance."
If thirteen yrch and their hobbit burglar had entered Fangorn, then the woods elves would have responded by killing all of them, immediately. The elves responded to the dwarves by imprisoning them indefinitely, and then later on, preparing to kill all of Thorin's dwarf allies in a big battle. So yes, there is a difference.