My BV6-26 was probably just as old (with an analogue tuner) and also had started life in a hotel. When using the DVI input my two problems were firstly getting the resolution right (as we have discussed), and secondly High-bandwidth Digital Content Protection (HDCP).
As far as I am aware, TV’s that are not HD-ready cannot display an hdmi signal with HDCP. You may find that you can go into your IdeaCentre’s graphics settings and turn off HDCP to see if this is the cause of the problem.
When I used an Apple TV with my BV6 I also used an audio extractor (to get the sound out of the hdmi) which had the advantage of stripping the HDCP from the signal so that the TV would display it.
I described my DVI experience in this post: https://forum.beoworld.org/forums/topic/hdmi-connection-for-beovision-7-mk1/#post-1707