I agree with you that automated requests can be slippery with unhappy customers. I considered that before I started to go automated, and came up with a template that I hoped would mitigate bad reviews and offer the customer an alternative to vent to me about a bad experience prior to giving the review. I can't say that this will work, but here is my template:
Thank you for choosing (insert dealership name) for the purchase of your ((Vehicle Type)).
I hope we provided you with great experience. If so, please visit this link and leave a Google review for us (insert google review link)
If we did not provide you with great experience, please let me know what we could have done differently to make your visit more pleasant.
If there is anything else that I can help you with, please let me know.
Thank you again. We appreciate your business and look forward to working with you again in the future.
(added my direct contact information)
I plan to give this automated message a run for a few months to see what happens, then go from there. If I am present at time of purchase, I do make an effort to personally ask for reviews before the customer leaves. This would be my preferred way, but I am not always nearby when the customer makes their purchase so I can't be consistent.