Agreed a CV is overkill. At most I wrote a polite letter to scores of places describing who I was, what situation I was in, etc. I got plenty of letters without mentioning predicted grades a CV, etc. For a lot of my placements I actually just called them and/or knocked on the door, most farms, kennels, etc are just happy for the free extra pair of hands.
You want to come across as a professional, nail your SPG, write formally, etc. You also want to come across as motivated, dedicated, hard working, etc. The last thing a practice wants is a dull student to stand in the corner all day with no interest in the subject yacking down his/her phone. Mention your previous placements, maybe enclose a few photocopies of your references, etc.