Just had a root about on Alan Applegate's site (K0BG.com) under "Wiring and grounding"
Here's a copy/paste from there
Using ignition-controlled relays to switch power off and on to amateur radio equipment is a common, but ill-advised, practice.
Rubbish. Alan and I disagree quite strongly on several aspects of his website when it comes to powering radios. I speak from a strong automotive and electronics engineering background. I'm not quite sure what his qualifications in this respect are but anyone who claims a 5W HT with no faults can draw so much current from a 12V AUX socket that it sets the car alight needs to have their advice treated with a large pinch of salt.
Please don't give any further advice on powering mobile installations.
Yours sincerely, a former vehicle mechanic and automotive electrician currently half way through a BEng (Hons) Electronics Engineering degree....
To the OP. What you are wanting to do will work just fine but make sure you have the correct rating of relay, I'd go for 20A just to make sure.
One word of advice though is before deciding to install there make sure there's plenty of ventilation. I don't know that dashboard but in many where the ashtray goes it is a solid recess rather than a hole in the facia so you would get a massive heat build up.