Can I take indomethacin if I have an extreme bout of gout even though I’m taking allopurinol...

There is a moderate drug interaction between indomethacin and meloxicam since both medications fall under the classification of nonsteroidal anti-inflammatories.

GENERALLY AVOID: Concomitant use of more than one nonsteroidal anti-inflammatory drug (NSAID) at a time may increase the potential for serious gastrointestinal toxicity including inflammation, bleeding, ulceration, and perforation of the esophagus, stomach, or intestines. These events can occur at any time during NSAID use, with or without warning symptoms. The risk is dependent on both dosage and duration of therapy. Patients with a prior history of peptic ulcer disease and/or GI bleeding have a greater than 10-fold increased risk of developing a GI bleed during NSAID use compared to patients without a history. Additional risk factors include old age, alcohol use, smoking, and poor general health status.

https://www.drugs.com/interactions-check.php?drug_list=1335-0,127-0,1549-0

Therefore, if you're still experiencing extreme symptoms of gout despite taking meloxicam as prescribed, it would be best that you talk with your doctor so he could determine if you need some dose adjustment or if there is a need to shift or add your medication regimen based on your health history and current health status.