Does your kettle have a mixing wing? That, combined with the immersion blender and straining thru a chinois should keep it lump free. I cook my roux right in the kettle, my milk goes in cold 10 liters at a shot, mixing with the wing the entire time. For example, roux cooked with wing running at 15 rmp. Add milk, 10 lte, close lid and speed up to 100 rpm with lid closed(safety messure) to incorporate milk, open lid and reduced wing speed to 15 rpm. Repeat.
I start and continue my seasoning(salt and white pepper) thru the process as in incorporates without pockets of pepper.
Reduce my heat and go to work with the robotcoup. Strain thus chinois. Into 20 liter bins and into the blast chiller