Change Azure Standard to Premium

The only way that I'm aware of would be to just start copying, robocopy is likely the most efficient method.

However, if you are aware writing backups to local disk uses up your alloted IOPS. It is best practice, and recommended by Microsoft, to use the backup to URL for database backups. Doing this does not count against your IOPS for the disk or VM.

If the indexes are fragmenting that quickly due to massive amounts of activity affecting arbitrary user-provided values then they may be little you can do.

If the indexes that are fragmenting badly are over UNIQUEIDENTIFIER columns then these will fragment quickly when they see a lot of activity because the data is effectively random. If this is the case then you can minimize this greatly by using NEWSEQUENTIALID() instead of NEWID(), or the equivalent in your application if you are generating UUIDs there instead of in the database.

If you are generating UUIDs outside SQL Server make sure that they are generated in a way that is sequential in SQL Server's sort order (different generators use different byte orderings) otherwise you lose some or all of the benefit. See this for reference to .Net's UuidCreateSequential not producing UUIDs with the same ordering properties to SQL Server's NEWSEQUENTIALID() and how to juggle the result to work.