Allow FastAPI 0.116 and Starlette 0.47

[musicinmybrain]

This includes the FastAPI update from #1014.

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 90.57%. Comparing base (17e0834) to head (89d3ce7).
⚠️ Report is 5 commits behind head on master.

Additional details and impacted files

@@           Coverage Diff           @@
##           master    #1027   +/-   ##
=======================================
  Coverage   90.57%   90.57%           
=======================================
  Files         150      150           
  Lines        4381     4381           
  Branches      534      534           
=======================================
  Hits         3968     3968           
  Misses        320      320           
  Partials       93       93           

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