Ball four gets lodged
 

NFHS - Ball Four gets by the catcher and lodges in the backstop. Does the batter get first base only or does he get first base on the walk and then second base for the lodged ball award?

The ball becomes dead. The B/R gets 1B and all other runners are awarded one base.

MTD, Sr.

Pitched ball out of play. One base for all runners from time of pitch.

Batter was not a runner at the TOP, he became a runner when the ball reached the plate.

Batter gets first from the walk. No award for the pitch going out of play.

Think of it this way, no count on the batter, pitch out of play. Move the runners up one base, batter stays at bat.

Yes, I know I'm just explaining what MTD said in my own convoluted way.

Your convoluted reasoning, which works in the OP, might not work when a DTK lodges in the catcher's gear when the batter could become a runner.

Perhaps so, but how is DTK lodged in player equipment any different. Same awards for the runners, one base from TOP, batter gets 1st base.

Actually, the FED dead ball table awards the batter 1B on the walk and 1B on the lodged ball. Except both awards are from his base at TOP so they are concurrent.
FED does not actually award the batter 1B on a lodged DTK but since he became a runner he gets 1B even though he was not a runner at TOP.