When I have the problem of not wishing to interrupt my site's operations, what I do is find out what the lowest period of traffic is during the day. There's lots of potential sources for this info, but the best is your site stats (if using cPanel, probably in Webalizer). Ask your host what kind of stats are available.
Then, I just do my work during the least trafficked period of the day. This may sound crazy, but I found that for about an hour or so during the middle of the night, the only visits to my site were by bots, giving me sufficient time to do my work. The bots don't care if the plugins are active, only that the site is, so I experienced no loss of SEO standing.
I'm not sure what Dragonfly does, but deactivating the plugins for a few minutes testing at an off-peak period would help immensely. You should also check the source of your plugins to make sure each is up to date with WordPress' current version. This helps alot, even if you can't deactivate all of them.
Meanwhile, I'd contact the developer of Dragonfly and ask them for their input.
I'm sorry to read of your troubles since upgrading. We're here to help with any issues as best we can. Please don't hesitate to open topics for your issues.