feat: Waitlist Lottery - Adjust Doorway Listing Type Filter

[millord]

This PR addresses #5506

• Addresses the issue in full
• Addresses only certain aspects of the issue

Description

Please include a summary of the change and which issue(s) is addressed.

How Can This Be Tested/Reviewed?

1. Go to the partners site and create two listing.

• open waitlist lottery
• available units

2. Go to the public site, select the filter and filter by open waitlist

• should see the open waitlist lottery listing you created

3. select the filter and filter by available units

• should see the available units listing you created

Provide instructions so we can review, including any needed configuration, and the test cases that need to be QAd.

Author Checklist:

• Added QA notes to the issue with applicable URLs
• Reviewed in a desktop view
• Reviewed in a mobile view
• Reviewed considering accessibility
• Added tests covering the changes
• Made corresponding changes to the documentation
• Ran yarn generate:client and/or created a migration when required

Review Process:

• Read and understand the issue
• Ensure the author has added QA notes
• Review the code itself from a style point of view
• Pull the changes down locally and test that the acceptance criteria is met
• Either (1) explicitly ask a clarifying question, (2) request changes, or (3) approve the PR, even if there are very small remaining changes, if you don't need to re-review after the updates

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[mcgarrye]

LGTM