Improved exceptional handler in unlocked message dispatch

[philippjfr]

Followup on #6443, ensuring that we handle exceptions correctly.

[codecov]

Codecov Report

Attention: Patch coverage is 0% with 16 lines in your changes are missing coverage. Please review.

Project coverage is 71.56%. Comparing base (0621ac3) to head (2e26f07).

Files	Patch %	Lines
panel/io/document.py	0.00%	16 Missing ⚠️

Additional details and impacted files

@@             Coverage Diff             @@
##             main    #6447       +/-   ##
===========================================
- Coverage   82.06%   71.56%   -10.50%     
===========================================
  Files         310      310               
  Lines       45929    45930        +1     
===========================================
- Hits        37691    32871     -4820     
- Misses       8238    13059     +4821     

Flag	Coverage Δ
ui-tests	?
unitexamples-tests	71.56% <0.00%> (-0.03%)	⬇️

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