Yes, you can test using F5 VIP as for MSSQL but for the mysql monitor so it can work with TLS traffic as per K40700745 .
Also see this article https://support.f5.com/csp/article/K15242 to maybe send smaller number of probes per connecton.
By F5 internal mysql client you mean "/usr/bin/mysql " that is used primary fo accessing the F5 local mysql database for AVR/ASM/APM statistics? If this is the case you can try making externel linux bash script monitor by using the sql client but it will be hard work and I hope the other previous idea with an F5 VIP helps. But just in case I will share this as examples that can be rewriten:
https://support.f5.com/csp/article/K71282813
https://devcentral.f5.com/s/articles/ltm-external-monitors-the-basics
https://support.f5.com/csp/article/K31435017