[SalesForce] Error while updating debug log trace flag

Realistically, all you can do is set the user trace flags to a debug level with all levels set to "none", then set the class's log to a debug level that you want. Yes, you'll still be generating a ton of logs, but the irrelevant logs will all show up as 1Kb, while the ones you want will be significantly larger. If you're using the Developer Console, you should even be able to click on the Size column header to sort by size; all of the ones you care about will be near the top/bottom of the list.

There are basically two options open to you:

1. Ensure that you set up a Debug Levels definition that minimizes most of the captured data, as covered in the documentation, whilst definitely capturing at least ERROR level for Apex. Indeed, for the issue you face you can probably set everything to just ERROR. Once you set up the debug levels, use it on the trace flags you create for the batch's execution user.
2. If you need more complete information you can still have more captured data if you can change the batch's scope chunk size, e.g. to 1. This means you will get a separate debug log per record processed by your batch. Depending how you wrote the code that calls Database.executeBatch this may require a small code change.

It may be that you could manually invoke the batch through anonymous apex, in which case you can simply set up the SFDC_Console debug level, open the developer console, access the Execute Anonymous Apex dialog and explicitly call the execute batch. The console debug level is automatically applied to the user you opened the developer console as.