Add test script for bypassing backend authentication issue

[b1ron]

Description

Refs #4115.

Readiness checklist

• I added/updated unit tests (and they pass).
• I added/updated integration/compatibility tests (and they pass).
• I added/updated comments and checked rendering.
• I made spot refactorings.
• I updated user documentation.
• I ran task all, and it passed.
• I ensured that PR title is good enough for the changelog.
• (for maintainers only) I set Reviewers (@FerretDB/core), Milestone (Next), Labels, Project and project's Sprint fields.
• I marked all done items in this checklist.

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 73.98%. Comparing base (89f22e4) to head (91c58d4).

Additional details and impacted files

@@            Coverage Diff             @@
##             main    #4119      +/-   ##
==========================================
- Coverage   75.11%   73.98%   -1.13%     
==========================================
  Files         339      339              
  Lines       22910    22910              
==========================================
- Hits        17209    16951     -258     
- Misses       4393     4661     +268     
+ Partials     1308     1298      -10     

see 28 files with indirect coverage changes

Flag	Coverage Δ
filter-false	?
filter-true	67.38% <ø> (-1.46%)	⬇️
hana-1	?
integration	67.38% <ø> (-1.54%)	⬇️
mongodb-1	5.13% <ø> (-0.01%)	⬇️
mysql-1	?
mysql-2	?
mysql-3	?
postgresql-1	46.49% <ø> (-0.09%)	⬇️
postgresql-2	49.65% <ø> (-0.12%)	⬇️
postgresql-3	49.81% <ø> (-0.14%)	⬇️
sqlite-1	45.66% <ø> (-0.08%)	⬇️
sqlite-2	48.73% <ø> (-0.13%)	⬇️
sqlite-3	48.97% <ø> (-0.08%)	⬇️
unit	32.69% <ø> (ø)

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