Hi Jonas,
> Firstly, if your custom logic means "Contacts/recipents could reset their password via a link in marketing email or by submitting marketing form",
the requirement need many back-end steps thus it couldn't be achieved only with front-end customization on Marketing email.
The available customization we could do is just embed URL which navgate recipents to reset password page in portal.
url.powerappsportals.com/.../ForgotPassword

> If your custom design means responsive columns layout such as Bootstrap or any CSS framework, just add their CDN as reference link into Marketing email HTML source code.
You could consider enable a early access for Dynamics 365 for Marketing, which add OOB columns layout feature.
https://docs.microsoft.com/en-us/dynamics365/customer-engagement/marketing/email-layouts
While making customization on Marketing email, be aware that marketing size is limit under 100KB to avoid to be flagged as spam.
(Refered images won't be calculated into size)
https://docs.microsoft.com/en-us/dynamics365/customer-engagement/marketing/email-design#design-your-content
You could embed specific attributes to your custom element to make them be recognized as draggable elements by CRM.
https://docs.microsoft.com/en-us/dynamics365/customer-engagement/marketing/custom-template-attributes
> Acutally, Send Reset Password Email to contacts and Marketing email are processed by different back-end engine,
the previous one is processed just as activity emails, it will be triggered after users submitted their email address in Forgot Password Page in Portal.
You can find this process in Advanced setting > Settings > Processes and edit its text content.(unfortunately it couln't be customized as Marketing email)

Please read following articles for details:
https://www.inogic.com/blog/2018/03/working-with-microsoft-dynamics-365-portal-password-reset/
https://meganvwalker.com/forgotten-password-functionality-dynamics-365/
while the latter has its own indepent engine.
Regards,
Clofly