I use RDP with my setup on my LAN without much issue. Sometimes the video playback, which I watch at 4x, gets jumpy but pausing the playback for 5-7 seconds seems to reset the issue. I pause until the static image blinks for an instant then playback is smooth again. This seems to happen more often with BI5 than it did with BI4 but I only moved to 5 this past weekend.
BI5 has their "remote management" option which you might want to check into. My understanding is that it requires an additional BI5 license if you wish to use it as the remote management console after the trial period expires but I could be wrong. You could try it out for the trial period and if it meets your needs perhaps it is one way to address the issue.
I would guess that if the BI PC is spiking to 100% cpu and this is a new behavior perhaps something has changed. Perhaps update your video drivers on your BI PC and your originating desktop. Check the temp folders on the BI PC and make sure they aren't full of left overs. You can find the two temp folders on the BI PC at:
C:\Users\<username>\AppData\Local\Temp
C:\Windows\Temp
If there is a bunch of stuff in those folders, more than 2 digits worth, you would want to reboot the BI PC then open those folders and delete everything inside them. There may be some files that cannot be deleted so you can just skip those. On very rare occasions I have seen software put files into the temp folders that you are able to delete but they are required for the software to work. This is very rare and is generally considered poor coding.
Another thing to consider is if the controlling machine has been changed to a higher resolution and you are running RDP in full screen. If the BI PC doesn't have a powerful video card and you are running a high resolution on your controlling PC the RDP session will have to redraw the BI PC desktop and apps at a much higher resolution than it may be easily capable of.
Perhaps open task manager with "more details" so as to see what process is burning all the CPU when you connect, it may help shed some light on what is causing this.
For remote access I use OpenVPN to my firewall then the BI web interface to the private IP or RDP to the private IP if I need to extract footage remotely which is very rare.