Is it bad practice to automate the updating of statistics and re-making of query plans on a relatively small SQL Azure database?
If auto update statistics is not enough for your DB, try a couple of things:


Manually update statistics - recommend Ola Hallengren's scripts that can be configured to maintain indexes and/ or statistics as required. Test different parameters (all/ indexes only/ sample rate etc) and test running daily/ weekly/ at start of proc and the like, and see what works best for you with least overhead.
Scripts are here: https://ola.hallengren.com/
Brent Ozar article about updating stats only with those scripts:https://www.brentozar.com/archive/2016/04/updating-statistics-ola-hallengrens-scripts/

Enable Trace Flag 2371 (if SQL Server version older than 2016)
For large tables, this reduces the threshold for automatically updating statistics dynamically. This algorithm is used as standard since SQL Server 2016.Good further reading:
https://support.microsoft.com/en-gb/help/2754171/controlling-autostat-auto-update-statistics-behavior-in-sql-server
https://littlekendra.com/2016/04/18/updating-statistics-in-sql-server-maintenance-answers/#trace-flag-2371