Let me first quickly note that /etc/X11/Xsession.d/ might be better than the desktop startup appications route that I mentioned above for the xrandr command. Am myself on Mint 17.3 and shall refrain from details which might not apply on Mint 18, but a reader may want to look into it.
Still stumbling about in the dark here but your providers 1 and 2 (i.e., your R7 265) are listed as supplying 0 outputs, the Intel one as supplying 3. If I'm not misinterpreting "outputs" that might just mean that your R7 does not itself have, well, outputs, but needs to feed the Intel card. That is, I believe we find ourselves in the "Discrete card as primary GPU" situation as described in that same Arch Linux article, https://wiki.archlinux.org/index.php/PRIME
So, does it work to create the there mentioned /etc/X11/xorg.conf.d/10-gpu.conf file, replacing every occurrence of "nouveau" with "radeon" and plugging in the correct PCI bus addresses? As to that latter part: if your Intel and AMD GPUs are listed as, say, 00:02:00 respectively 01:00:00 in the output of
you'd use the in that example used specifiers "PCI:0:2:0" respectively "PCI:1:0:0".
Logging out and back in is probably enough also on Mint 18 (and with LightDM as on 18.2) but if not; rebooting certainly is enough to have the changed X configuration active. Careful. If the above is incorrect you may find yourself with a non-starting X server. If so and if not already there, switch to a virtual console with Ctrl-Alt-F1 or -F2, login as your user,
sudo rm /etc/X11/xorg.conf.d/10-gpu.conf
Also note the bit about
from that artile if you find it to be necessary; still use 0 instead of "Intel" and 1 or 2 instead of "nouveau".
Seems promising. But, well, if not, then I do believe you may want to wait for someone with actual hands-on experience here. Full VGA switcheroo, as far as still applicable in the first place, is a bit of a disaster when you don't have it locally to test things.