How can I help my career with a broadening assignment?

First, you can't PCS again within a year.
While working at TRADOC isn't a broadening assignment, it is considered a more challenging assignment and is viewed more favorably than working as a SL in a BCT. You can submit for a broadening assignment at one year on station, but you are filling a demanding assignment already.

Branch is correct, you won’t be able to move if you’re less than 12 months on station. Also you have to keep in mind unless you get selected for SFC, Recruiter or Drill Sergeant, you’ll be on station for 36 months. While assigned in your current position there are additional duties you can attend training for to help with your career (EOL, Victim Advocate/SHARP and MRT. Good luck to you....

Being stagnant in the same position for a long period of time may or may not hurt you. If you are already KD complete then yes, I would try to get something different. A broadening (drill, recruiters, ext..) is a good next logical move but if you just got to a place and branch is saying you don’t have enough time on station, it likely will not happen. A few things you could do in the mean time to help yourself stand out is college, attend some schools, and potentially work into a staff position relative to your MOS. While it’s not a broadening assignment it does look good getting rated time outside of your normal day to day MOS.