Hi,
I’m not clear why you can’t make the button look the way you want it to in your plugin and keep it that way. Why must it be different based on theme or different browsers?
The buttons in your site are behaving like that when viewing the forms in different browsers because of the current Twenty Eleven theme. That is why I suggested for you to search for a different theme. One more advanced, like Twenty Sixteen and above. However I understand your current predicament.
The following documentation should help you customize the login widget.
https://simple-membership-plugin.com/customizing-the-look-of-the-membership-login-widget/
The registration is a bit different. Here is a the current class used.
.swpm-registration-submit
You should be able to use the above class to customize the button.
Let me know if the above helps you.
Kind regards