If the site is less aggressively blocking but only has a per-IP rate limit, buy a subscription to one of those VPNs (it doesn't matter if they're "actually secure" or not - you can borrow their IP addresses either way). If the site is extremely aggressive, you can outsource to the slightly grey market for residential proxy services - for fifty cents to several dollars per gigabyte, so make sure that fits in your business plan.
There's an upper bound to a website's aggressiveness in blocking, before they lose all their users, which tops out below how aggressive you can be in buying a whole bunch of SIM cards, pointing a directional antenna at McDonald's, or staying a night at every hotel in the area to learn their wi-fi passwords.