Update on this. I called the credit card company and talked to three different people at Chase. The response I got was the same:

Person at Chase: "We won't stop you at your credit limit, and we'll let you go above it, but we can't say exactly when we will stop you. But we could do a hard pull and request a credit limit increase."

Me: "My current credit limit is $20k and I want to put a $250k transaction through."

Person at Chase: "Oh, yeah we can't help with that."

Me: "Do you offer any additional products that might be able to accommodate this?"

Chase: "No"

Me: So I should just get an AmEx even though I'm trying to funnel a bunch of expenses through your company?

Chase: "Yes"