I’m wondering if my 1.1 Ghz Pentium M on my TR3AP can handle the job of a 1.5 P4 as required by the following TV tuner card:

http://www.aver.com/products/tvtuner_avertv_cardbus.shtml

Or has anyone used this or a similar product?

Generally speaking, a Pentium M is 800-900MHz faster than an equivalently clocked Pentium 4. Thus, a 1.1GHz Pentium M performs about the same as a 1.9-2.0GHz Pentium 4.

You should be ok.