Composite and SCART are analogue so not Full HD..
SCART is not HD at all. It's standard definition.
If you have a set of 3 phono sockets on the back of the TV that are Red, Green and Blue, like this:

along with a white one and a red one (for audio), then that will be an Analogue HD input.
But the TiVo box doesn't have analogue HD out, just HDMI for HD or SCART for standard definition.
See

More importantly it sounds like your LCD TV can't do HD at all.
It should say "HD Ready" on it somewhere if it can do HD.
If you don't have either an HDMI, DVI or a component connection like above with the 3 phono sockets, then it can't do HD.
Not all LCDs can do HD.
If that's the case, you'll need a new TV to view TV programmes in HD and until then you'll just have to use it on SCART and watch in non-HD.
As for your bedroom, does the Asus monitor have an HDMI socket like this (the circled one):

Then that's HDMI, and you can just run an single HDMI cable from the TiVo box to the monitor.
If not, and it only has a DVI, then you'll need an HDMI to DVI cable but then you'll also need to run another cable for sound as DVI doesn't include sound.
HTH