I did it the following way.
My Commercial was a Multi initial. About 100 hours towards the 120 hour total requirement, I ended up flying from the right seat with my MEI and worked on the Commercial manuevers, and teaching.
The last 5 hours of the 120, I went back to the left seat, made sure I was still performing above standards so I could knock the checkride out of the way. Sure enough, good as green - and checkride completed. CMEL in hand. Same day, I also took my CSEL ride (as I had also been flying some SE stuff during those 120 hours).
All 141, and the VA paid for it. Now, by being extremely pro-active, I knew how much money I would end up getting back from the VA. It ended up being $5000 more than what I ended up paying for the training I had already completed to that point. (I had a 20k loan that took me from Instrument airplane, through to CMEL and a good portion of CSEL), and ended up flying on credit with the school I was training at since they had a few issues with the VA and my paperwork. In the end, I racked up 14k on account. I got 19k back from the VA after the paperwork was finally filed.
That 20k + 14k gave me a grand total of 34k spent on my training post-PPL.
I soon found out what the CFI 141 minimums were, something outrageous (I'd look it up. . .but it was what - 15 or 35 more hours of additional flight time?)
Hell no. . .sorry. . .I'm already proficient why the hell am I going to go into MORE debt, just for that when I can do it all Part 61.
In the end, after my CMEL checkride, I did 3 more hours of twin flying in prep for my MEI.
Those 3 hours cost me just shy of a grand. Plus a 1.5 hour flight, my MEI (Initial CFI) cost right at a grand.
Part 61. $4k left to do my CFI SE-Addon. 10 more hours in the right seat of our single trainer, plus a checkride - about $1200.
Leaving $2800 to throw towards the loan.
Just think about it, do your math, and see if doing your CFI's Part 141 will really leave you in a better financial position.
If you haven't figured it out yet, I made $2800 by doing all of my instructor ratings Part 61.