GTA IV on Xbox 360 – Out of Commission – Helicopter bug

I’ve hit on an Xbox 360 the same bug described in GTA IV – Out of Commission – Helicopter bug:
I am on the last mission of GTA 4 — Out of Commission — and cannot finish due to the bug with getting on the helicopter. The pop-up instructs me to repeatedly hit A, however A is mapped to “run” (and NOT to “jump/climb”). So it never works.

The solutions proposed in the linked Question do not seem to apply to the Xbox platform:

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    • I cannot use a wired keyboard
    • I couldn’t find a way to remap the jump key
    • I couldn’t find a way to set the fps

    So how can I work around this bug on an Xbox 360 so as to complete the said mission?

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