Can I get anything other than a honorably discharge while being flagged for height and weight during my ETS process?

What can you do to fight the flag? Nothing, unless you can show that there is an underlying medical condition that is causing you to gain weight. As the ABCP regulation* says, "The inability to exercise does not directly cause weight gain." In other words, if you can't do push-ups, then you need to do more push-aways.

However, the type of discharge* determines the characterization - not if you are flagged at the time of your separation. If you are a "Chapter 4" discharge (separation for expiration of service obligation), then the only characterization allowed is honorable.

You said "may be flagged", so I'm assuming you aren't currently enrolled in your unit's weight control program, correct? If you're not a 'repeat offender', then the ABCP process runs for a year and then you would be possibly separated as a "Chapter 18" discharge (failure to meet body composition standards). However, even then, the characterization of service for a Chapter 18 discharge would still be honorable (assuming there is nothing else that would affect it).
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* AR 600-9 (The Army Body Composition Program) - https://armypubs.army.mil/epubs/DR_pubs/DR_a/pdf/web/ARN7779_AR600-9_FINAL.pdf
* AR 635-200 (Active Duty Enlisted Administrative Separations) - https://armypubs.army.mil/epubs/DR_pubs/DR_a/ARN40058-AR_635-200-001-WEB-3.pdf

I've not seeing anyone get less than Honorable for HT/WT chapter. Notifying someone could get less, but I've never seen it. But you're asking about regular ETS while flagged for HT/WT. If you get less than Honorable, I'd be surprised

See below from AR 635-200 Ch 18. However, if you're already ETSing and you only have a few months left, even if you get flagged, you'll still get an honorable discharge. Have you even been flagged? Because you said you "may" be flagged before you ETS. Do you even think you'll do ht/wt if the unit has one before you ETS? Even if they do, you wouldn't even be able to complete ABCP. But it says in the reg you get an honorable discharge even with a flag for weight.

18–2. Criteria
a. In accordance with AR 600–9, a Soldier who fails to make satisfactory progress in the ABCP or exceeds the body fat standards within 12 months after release from the ABCP, and the required medical evaluation (see paragraph 18–3) finds no underlying medical condition, will be initiated for separation. Commanders have the discretion to bar Soldiers from reenlistment in lieu of initiation of separation per AR 601–280.
b. In accordance with AR 600–9, a Soldier who, after 12 months but less than 36 months from the date of release from the ABCP, again exceeds the body fat standard and is not found to have a temporary medical condition that prevents weight or body fat loss will be re-enrolled in the ABCP. Soldiers who fail to meet the ABCP body fat standards within 90 days from reenrollment will be initiated for separation. Commanders have the discretion to bar Soldiers
from reenlistment in lieu of initiation of separation per AR 601–280.

18–3. Procedures
a. Prior to the initiation of separation under paragraph 18–2, a medical evaluation is required. The healthcare provider will ensure the Soldier can participate in the ABCP and rule out any underlying medical condition that may be a direct cause of significant weight gain or directly inhibit weight or body fat loss.
(1) If no underlying medical condition exists, the Soldier will be processed per paragraph 18–2.
(2) If the exam determines that a medical condition exists that does not meet medical retention standards of AR 40–501, the healthcare provider will refer the Soldier to an MEB.
(3) If the Soldier is found to have a temporary medical condition that directly caused weight gain or prevented weight or body fat loss, the Soldier will have up to 6 months from the initial medical evaluation date to undergo treatment to resolve the medical condition. The medical specialty physician may extend the time period up to 12 months if it is determined more time is needed to resolve the medical condition. Once the medical condition is resolved or 6 months (not to exceed 12 months), whichever occurs first, from the date of the medical evaluation, and the Soldier still exceeds the body fat standard, he or she will continue participating in the ABCP and will be required to show satisfactory progress. A Soldier who fails to show satisfactory progress will be processed per paragraph 18–2.
b. Prior to the initiation of separation under paragraph 18–2, commanders will verify that all rehabilitative and counseling requirements of AR 600–9 have been met.
c. The notification procedure will be used for separation under this paragraph (see chap 2, sec I).
d. The provisions of chapter 1, section VII will govern whether the Soldier will be released from active duty with transfer to IRR or be discharged. Refer to paragraph 1–12 for additional instructions on ARNGUS and USAR personnel.

18–4. Characterization of service
The service of Soldiers separated under this chapter will be characterized as honorable or an uncharacterized description of service if in entry-level status. The commanders specified in paragraph 1–20 are authorized to take final action in cases processed under this chapter.