On some, but not all posts, in the process of rebuilding, I receive the following message:

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Out of memory during "large" request for 1052672 bytes, total sbrk() is 21212120 bytes at lib/MT/FileMgr/Local.pm line 131.
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When rebuilding the same posts later (after edits, etc), the error continues to happen, even though other posts have no problems whatsoever.

Please advise.

You may want to check with your host and see if they have imposed any sort of limit on how much memory a process can use.