I withdrew my under $3000 401k due to a sudden job loss. Can I select "eligible emergency expenses" or "other" to possible reduce my tax bill

Yes, sounds like it may qualify as an "emergency." An emergency will allow you to make a withdrawal, still subject to tax, but without the 10% early withdrawal penalty.  You can take one financial withdrawal emergency per year, not to exceed $1,000. So, $2,000 of your withdrawal won't be eligible for penalty relief.

Here is some more information: What are the exceptions to the penalty for an early withdrawal from my 401(k)?

"Other" does not apply.

$1,000 of the $3,000 would qualify for the emergency-expense exception only if you actually needed the money for an unforeseeable or immediate financial need.  Losing a job would not necessarily create such an financial need.

See Q&A 1 and 2 in IRS Notice 2024-55:  https://www.irs.gov/pub/irs-drop/n-24-55.pdf