Genghis,
Your IR is easy, written test and training to proficiency which will probably be 4-5 flights to get familiar with all the local approaches the examiner is likely to ask of you.
It would make sense to combine the training for the CPL and CFI, no sense in doing the same thing twice from different seats.
There is no requirement to fly the CPL checkride from the LH seat so use it as a practice run for your initial CFI. Find a school that is willing to do this and that has an instructor available that is experienced enough to make it work.
I’ve done this successfully before maybe 3-4 times.