Controls for BYOD

The issue is that not all employees have been granted a company-sanctioned device for accessing our office’s network. Policy-wise we have restricted them to only using RDP and that they must have anti-virus software on the machine they’re using to access our network, however, I feel this is not enough. I do not have as much experience in this area as others may have, so I’d like to know how do other companies deal with such a thing? Can we force our employees using their own devices to install an MDM software provided by the company so that we have some control over what’s connecting to our network? Is that the best route or will backlash/pushback from the employees being forced to install something they may deem intrusive actually work against us in this process? What about employees accessing our email system (office 365) on their personal phones – should we extend MDM and controls to those devices as well? Can/should we define a list of acceptable phones/operating systems to be used in our BYOD policy?

Any insight you could provide here would be greatly appreciated. In a perfect world, I would issue each employee a company-owned laptop/phone to deal with this situation, however, we just do not have the budget to go that route.