On the SQL Server, try to modify Database Properties for the AX database and change Recovery model to "Simple" (if it's currently set to "Full", which seems to be the case with transaction log eating up the disk space). SysDatabaseTransDelete() class can be really slow, even on a fast and optimized SQL Server. You can also try to delete records from the largest transactional tables directly on a SQL Server, and after that run SysDatabaseTransDelete() to clean up the rest. In this case, take care to use DataAreaId parameter on T-SQL delete statement, so you don't delete transactions from other companies.
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