In reply to:
Unfortunatly, the system will be for/used by other companies, so I can't control who they put on the lists.
Then my bet is you're not going to be hosted here for very long unless you process your mail elsewhere. On a sheerly personal note, I would add that I don't think much of your business model in that regard either, but not knowing the details may or may not mitigate that opinion.
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I can filter out bounced emails
That sort of listwashing won't help the potential spamming problem at all.
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but can't control opt-in/out unless the client wants us to manage that.
Then that's precisely what you should do; do not offer service to any company that will not allow you to manage the process. Your customers will get better service and you will get more billable hours out of the process, and most importantly, the rest of us may get less spam.
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How do the other many email campaign companies get around this problem?
Many don't; they use proxies in foreign countries and/or zombie-bot networks to send spam.
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Own their own mail server?
In some cases, but most hide behind proxies.
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Any ideas of who to use?
Yes, but I'm certainly not going to post them here. Have a look at the SBL list on spamhaus.org and pick one of the many blacklisted domains offering that sort of unrestricted bulk email and select away.