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And crucially ensuring that beam continues to run for billions of years. a few decades may pass for you, but not for the laser that's pushing you

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Not sure if my interpretation here makes any sense but here it is. Due to relativity time slows down on the ship, the laser is still hitting the ship, how would the measurement of the energy received from the laser be perceived on the ship, from the ships perspective a 1000 years worth of energy arrives within a few decades. My head hurts thing about this. If you had a ship that could instantly jump to 0.5c travelling away from earth and we beamed a message using a laser at say 1Gb/s, how would the bandwidth appear to the ship?

You have it backwards. The further and faster the ship goes, the less energy is received. The energy collected over a day will be spread over a week, a month, a year, a decade, etc.

GP has it backwards too. For 1g of acceleration, none of the beam that is produced after a year will ever reach the accelerating spacecraft. You only need to run it for a year.

For 1g of acceleration, none of the light produced after 1 year will ever reach the spacecraft. You only have to operate the power source for a year, but you only get a year of source power spread across an infinity of time.

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