How do I repair an external hard drive with a corrupted catalog?

DiskWarrior is quite probably the only tool that could fix the drive non-destructively. Everything else is going to just be to scavenge what it can find of your data, not fix the drive. Testdisk is probably the only free tool worth looking at.

If the disk is not mounting, then you will likely have to spend some money on a repair utility or a data recovery application, if the data is important and has no backup.

Although you have specified that you are not interested in commercial software, I list below some such tools, if no better answer comes up. You may ask around your friends or nearby repair-shops that might have one of the below-mentioned products, which you may use freely or at least cheaply. These products have demo versions that you can try to see how effective they would be, to see if they are worth searching for (or buying).

Stellar Data Recovery has a demo version that will recover up to 1 GB of lost or deleted data, free of cost. The full version will cost €79.

Data Rescue by Prosoft Engineering has straight recovery options, but also advanced options to try to recover data from disks with physical failures. It has a demo version and has an interesting pricing model, where the price is determined by the amount of data that you wish to recover, starting from as low as $19.

DiskWarrior has a data recovery type option - it will try to rebuild and present a preview of what it thinks should be on the disk even if it can’t repair, and it might allow you to copy the data off. It is very costly at $119.95.

EaseUS Data Recovery Wizard for Mac is a subscription product, starting from $89.95 per year.