“Inconvenience” would be the verb for causing an inconvenience. So in the sentence you’re going for, “inconvene” would have to be replaced with the passive “be inconvenienced” (“we’ve gotta be inconvenienced and grovel to google a bit”). I don’t believe we have a separate word for “endure an inconvenience”, although it seems like the kind of thing some languages might have a single word for. Stylistically I’d probably restructure the sentence to “we’ve gotta put up with the inconvenience” rather than just using the passive verb, but yeah.

I think you’d most often see this verb in the stock phrase “Sorry to inconvenience you”.