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If this is your "local USPS branch", then it would pay good dividends to develop a relationship with the clerks. Then when you have an unusual request, they are likely to accommodate you without any hassle.

As to the post card, there are internal directives to not cancel machinable mail at counters to save time. That is the default decision for window clerks are
requested to follow..
As John Becker says, once you do get the hand cancellation there is a large chance it will be machine sprayed cancelled during its time in the mail stream. Now a read of the regulations indicates the clerks can hand cancel items of a philatelic nature but also have the right to refuse to do it on demand. They then need to set a time for you to return to get you requested hand cancellation. If after that, you don't want a machine cancel, you can pay the non-machinable fee, and mark the item as being non-machinable. You can also just mark the item as non-machinable with the addition fee paid and point that out to the clerk for the appropriate hand cancel.
Again, having a relationship with the clerks will help smooth issues over.
The tax, ballot return and certain other envelopes are hand cancelled upon request as such items are understood as being post marked based for certain qualification. Yet I have had occasion where large offices have refused to counter cancel even when requested.When I know I am dealing with such an office, I short pay the postage by one penny and have the clear affix a one cent meter strip. One clerk smiled at me, too my penny and quick as a flash place a one cent stamp on the item. Wait a dramatic pause with the smile then hand stamped the envelope with a round dater.