It was a communications error on my part Chipwizard. I assumed "thin 0.25 inch" was the HEIGHT of the PCB, where it seems you meant the WIDTH was 0.25 inches.
If you have to make such narrow width PCBs full of LEDs (do they go in a tube??), maybe you could make the LEDs surface mount, and TSSOP shift register (or whatever small package you like) on the rear of the PCB. A through-hole double sided board will allow some vias under the IC and a few leadouts each side but it's still going to be tight as the IC requires 3 control wires, 2 power wires and 8 output wires...
Why on earth do you need to make matrixed individually-controllable LEDs in a tiny "stick" form factor??
A lot of design problems are best fixed at the beginning... What are the LEDs displaying, how many LEDs total are required, what is the total overall size of the assembly and and what might be the best way to do that task?