Yes, I work in China facing these challenges constantly. Non-locally-hosted doesn't make sense. Locally host everything. The issue isn't only availability, it is speed. Host in China. You'll probably need an ICP License for this (actually, for anything online), so make sure your customer gets you one (as they're legally accountable for what they agree to host in China, as they should be, and are, the 'legal person'). Alibaba have VPS solutions very similar to Linode or DO. Get your customer to agree and pay for this, as you are legally not able to.
You mention accessing US sites: Is this front end or back end? Locally hosted servers face the same restrictions accessing outside the mainland as home connections do unless specifically negotiated.
Do have mobile, at least as a responsive level. I have no idea who your customer is, but it sounds like they're in China and are not-so-tech. That means they will likely check-up via whatever's convenient, and that would probably be via mobile.