It is not so much the degree type that is important, as you mentioned it is getting into a 6-12 month program, graduating from that program, and then passing your boards to get your ASCP certification.
You can become ASCP certified with any science related bachelors degree...(Bio, Chem etc) You do not need a BS MT/CLS degree, so as of right now you are all set to get into a program. So if I were you I would start applying to local MT schools at hospitals or even going back to school for a semester or two and loading up on Hematology, BB, Urinalysis classes in order to boost your chance at getting a spot. Maybe even apply for a job as a lab aide, or specimen processor to get some lab experience to give you an edge over the other applicants.
As far as your alternate idea of becoming MLS(ASCP) certified...
I believe the ASCP does have a route where you can go MLT(ASCP) -> MLS(ASCP) as long as you have a bachelors degree and a couple of years of lab experience.
Here is the problem for you going that route...
1) In order to begin working as an MLT in your quest to get experience you will need to become MLT(ASCP) first to even get hired as an MLT. Most MLT's completed a 2 year associates degree with/and a 6 month hospital program. Even though you got the hard part over with and got a bachelors, I think you will have a very hard time even getting ASCP certified as an MLT without completing an MLT program, and those program spots are probably reserved for students in the 2 year program.
2)If you do somehow figure out a way to be hired as an MLT and start getting your experience I have heard stories of the ASCP only allowing you to sit for the portion of the MT exam you have your MLT experience in.
Example...If you work as an MLT in Hematology for 5 years then you can sit for HH (ASCP) certification but you will not be allowed to be MLS(ASCP) certified in Chemistry/BB/Micro etc. (anyone confirm this?)
If I were you I would just go the MT school route.