fix(vip-search): PHP 8.2 deprecation notices for dynamic properties

[sjinks]

Description

This PR fixes deprecation notices in our codebase for the Enterprise Search.

Changelog Description

Plugin Updated: Enterprise Search

Fix PHP 8.2 deprecation notices

Pre-review checklist

Please make sure the items below have been covered before requesting a review:

• This change works and has been tested locally (or has an appropriate fallback).
• This change works and has been tested on a Go sandbox.
• This change has relevant unit tests (if applicable).
• This change uses a rollout method to ease with deployment (if applicable - especially for large scale actions that require writes).
• This change has relevant documentation additions / updates (if applicable).
• I've created a changelog description that aligns with the provided examples.

Pre-deploy checklist

• VIP staff: Ensure any alerts added/updated conform to internal standards (see internal documentation).

Steps to Test

CI should pass.

[sjinks]

WP nightly CI jobs may fail until #4903 is merged.

[codecov]

Codecov Report

Merging #4904 (e68534f) into develop (8761eba) will not change coverage.
The diff coverage is 0.00%.

@@            Coverage Diff             @@
##             develop    #4904   +/-   ##
==========================================
  Coverage      29.22%   29.22%           
  Complexity      4723     4723           
==========================================
  Files            277      277           
  Lines          20816    20816           
==========================================
  Hits            6083     6083           
  Misses         14733    14733           

Files	Coverage Δ
search/includes/classes/class-health.php	55.97% <ø> (ø)
search/includes/classes/class-healthjob.php	22.41% <ø> (ø)
search/includes/classes/class-queue.php	83.99% <ø> (ø)
search/includes/classes/class-search.php	82.58% <ø> (ø)
...earch/includes/classes/class-settingshealthjob.php	34.22% <ø> (ø)
...ch/includes/classes/class-versioningcleanupjob.php	48.43% <ø> (ø)
...ncludes/classes/commands/class-documentcommand.php	0.00% <0.00%> (ø)