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You need the same bankroll, it is all one long session.

The bankroll requirements are so that you can "weather" the ups and downs that the variance of the game hands you, so you don't go broke before you achieve your +EV edge in the game results.
If you play part time, the up and down streaks will occur over longer stretches of time, and it will take longer to achieve your final results, or smooth out the hills and valleys. If you play full time, part time, continiously, or one hand a day, the same ups and downs, streaks and runs, are there -- it just occurs over different periods of time.

It also depends on how you define your bankroll. The 300 BB bankroll is for full time play with nothing but your play being added to it. You could start with a smaller bankroll for part time play, if you are willing to add to it from your other income, in case your initial bankroll gets low (what you are calling starting bankroll really isn't your bankroll in the strict sense of the word).