In order to have stability, you will absolutely need a stable temperature.
you should use a TEC for this with a suitable controller. If you try this, you should get a lot closer to your needs.
Using a TEC to control or keep temperature stable will add unnecessary complexity, depending on the laser package, mechanical and thermal constraints, etc.
You have not disclosed any details about the Photodiode that monitors laser optical power. If it is integral to the laser package then it’s likely your APC control loop does not track properly. External PD monitoring is problematic and can contribute to inadequate power tracking.
Ideally the optical power will be captured by the PD and accurately fedback to the error circuitry in a linear fashion so proper tracking can be achieved. In the real world things sometimes don’t work as they should because there are design considerations that are easily overlooked or assumed, like self heating, thermal package drift, CTE errors, etc
If you want some additional help, please provide specifics about your design,.....ie: APC schematic, PD config, physical setup, etc
We can probably help you better after you consider PD loop error sources
@grouchobyte