Updating your "Limit product sales to:" in-stock inventory figure does not change the button code we issue, so your existing button codes for that product that you already pasted into your page should continue to work fine.
The only product settings which alter the button code are Item Number, Variants/Variations, and Suggested Price if you are using "Let buyers edit price", so only if you change those settings would you need to recopy new codes and replace the code on your site. Editing any other settings should not affect the button codes, so the buttons already on your site should continue working fine.