First, reject the label. Calling yourself Hispanic is like calling yourself an "English speaker". It's meaningful only to people who want to foster identity politics so they can ignore individuals. I have relatives from Puerto Rico, and relatives from Mexico. They acknowledge that they have very little in common.
Then when you turn to the Republican community you have to face comments like those above, making you feel like you don't belong there either.
There are nativists and xenophobes of both persuasions. I think the major distinction is that racism or group identity politics is an accidental characteristic of some conservatives, whereas it is an actual policy of the left. The liberals mask theirs with condescension. You belong where the truth is.
Three of the most powerful people in the country are black, and they are Republicans. There is no black figure of any real power in the supposedly "tolerant" party. The first Hispanic on the US Supreme court will be Republican--if the Democrats don't block him. I'll let you judge where the really bigotted people are for yourself--on the basis of what they do.