Ben, welcome to the club! We have been wrestling this issue ever since ESRI converted the initially wide open free service to a subscription based product.
So, you have clients with sensitive mapping info they don't necessarily want open to public access. Pretty normal. We do too. And to get them to go online themselves, set up an ESRI account, buy the minimum 5 seat package for $2500, get all that set up and running, ... well, pretty hard to make all that happen. But that is the business plan for the product. It eliminate most small client applications.
The good news is that there are some other ways to get them set up. If the client has some existing ESRI licenses, the get one named user for ArcGIS Online for each ArcMap seat they own. Most private clients won't have GIS software (that's why they hired you, right?), but some do, and many agency clients do as well. So those people become your target clients, and there is not much you can do for the small client, unless they write the $2500 check. If your map is important enough to them, you could get them to buy a single seat of the basic ArcMap and get the named user account that way, but the licensing doesn't allow for multiple users to use that single seat. And a single seat in even a small company doesn't allow everyone there to benefit from the online mapping. Even for in-house use, you can quickly reach the license limit for your AGOL licenses.
Also, be aware that when publishing data out for others to consume, you can expose your published services for others to consume, if not done properly. Be sure you get some expert help to be sure your client's data is not out there for someone else to snag.
Hope that is helpful.