The DMCA Safe Harbor provision only protects a hosting provider who "responds expeditiously to
remove, or disable access to, the material that is claimed
to be infringing". I don't think there's much leeway for the hosting provider to give you a chance to file counter-notice before they take it down. So you're left with trying to convince the hosting provider to forgo Safe Harbor protections, which means they would have to be really confident that you could win easily, quickly, and cheaply.
You can try to preempt things by hosting everything yourself, but they can always go after your ISP. Unless you own the whole internet[1], there's somebody who's not you who can be blackmailed into disabling access to your content.
([1] or otherwise render the rest of the internet ineligible for safe harbor, such as by making them financially benefit in a way directly attributable to whatever activity may be alleged to be infringing.)
You can try to preempt things by hosting everything yourself, but they can always go after your ISP. Unless you own the whole internet[1], there's somebody who's not you who can be blackmailed into disabling access to your content.
([1] or otherwise render the rest of the internet ineligible for safe harbor, such as by making them financially benefit in a way directly attributable to whatever activity may be alleged to be infringing.)