If this is a desktop computer that will not be leaving the house, will always connect to the home internet connection. Set your router's DNS to the OpenDNS service and set up an account. You can then get monitoring reports of each and every website they've been to and select levels of filtering if you want.
Every computer on your network will use the DNS addresses specified by the router, and the filtering will be completely transparent to the kids.
If you want to see what they're typing on those sites, you'd need a keylogger or parental control software of some sort. The most important thing there is that your kids only be able to use limited user accounts and that you don't ever give them the password to the admin account with which you'd control the computer.
Regarding the monitoring software coexisting with your virus software: You shouldn't try to find a tool that can hide from the AV, you should find a tool that's legitimate and thus can coexist with the AV software without raising alarms. Most AV software allow you to choose applications to "whitelist" so that even if the software behaves in a way the AV software thinks is suspicious, it will leave it alone.
Making a house rule that all computers are always kept in the living room or dining room or den or other open and public room that everybody has access to all the time can be as good a filter and monitor as anything else.