I have exactly those 2 products and the LEDs blink quite nicely. I have a blinker installed on my 41 switcher and a CP Fire Fighter car. As long as I don't think it'll kill me or cost me a lot of money, I connect stuff and worry about the ramifications later.
I did connect the lights in series. I had done that on a previous project mistakenly and had to take it out and redo it in parallel.
Sorry, the lighting regulator will be for the LED light strip in the ceiling only, I will have a separate circuit for the blinking LED. I hope to be able to position the single blinking LED in front of the rod to light up both sides. It'll be a couple of weeks before I can get to that project though.