How to become a medical technologist/clinical laboratory scientist...?

Since you live in the Houston area, you are in luck, call Hermann and ask for the path lab.  Ask to speak to a supervisor and pose your question.  They will be familiar with all the options in the area.  I think probably UTHSC Houston, and maybe even Baylor have programs, if not more.

Back when (I did that), you applied after 3 years biology major and received a bachelor's degree when you successfully finished the one year intensive training.  That has been over 30 years ago, so I am reluctant to give you specific advice, only options for finding out.

When I did that, you got your bachelors in bio /micro/etc and the did a one year intense training where you rotated through out all the hospital lab depts.  I would skip the masters for now; it is not needed and its would add2-3 years onto your education, and you might forget some things you need for med tech.  Get the med tech and then you can work on the masters later and possibly get some tuition reimbursement.

Now you can still do it the 4 + 1 way as well as the all-included bachelors route.

Major in CLS and attend an accredited med tech program so that you are eligible to take the certification exam. Just getting a BS in any science major will not be enough to be able to take the board exam.