Hello @JReekes and thanks for writing in the community.
It is not possible to mess with the specific module because it contains hubl tags that automatically create the output, just like a cta works or some other hubl tags, they generate the output internally so you can’t touch it.
You can see some info here in this link regarding the tag.
Now your best choice is to try and replicate this module in a custom module using the output that the default module produces. So this is split into two parts.
1. Recreate texts and layout:
This part is easy you just add a field in your new module that outputs text, then you put that text in whatever tag you like.
As a side note I would suggest you add rich text so you get full control over what is shown and it is furure-proof as you can always add more markup and elements, maybe a subheader or some explanation text with links or something, maybe some emojis or icons, or even images.
2. Recreate the form:
First add a field of form to your module.
This is a bit trickier, you will have to use javascript to produce a form that does the subscription, you can find a lot of details in this link where it guides you through the form creation, this is particulary useful when you need a copletly custom form or you want your hubspot form outside of hubspot, a differrent usecase.
So with number one and two, this is how it should look like:
<p>{{ module.title }}</p>
<div id="hs_form_target_blog_subscription"></div>
<script charset="utf-8" src="//js.hsforms.net/forms/current.js"></script>
<script>
hbspt.forms.create({
portalId: `{{hub_id}}`,
formId: `{{ module.form_field.form_id }}`,
formInstanceId: '60',
pageId: `{{content_id}}`,
pageName: `{{content.name}}`,
redirectUrl: `{{module.form_field.redirect_url}}`,
css: '',
target: '#hs_form_target_blog_subscription',
formData: { cssClass: 'hs-form stacked' }
});
</script>
The whole thing in your design manager should look like this:
Please don’t hesitate to ask for any further help, I know this info seems a bit overwhelming, especially the part where the form is created.
Always look up the docs, it is a wonderful place! 
If my answer was any helpful, consider marking it as a solution.