Image Captcha Refresh not displaying refresh link
I have tried recreating your problem and i saw that after you enable the refresh module for the image recaptcha, a link is added below the image in all the forms where captcha is enabled.

But there is no configuration page specific to this refresh module. The link works fine in reloading the image. 

I suggest that you add your form via configuration page of the captcha module instead of adding it in the code.

just include your form machine name and captcha in the captcha module configuration page.