Hey Christian, Other than the apps you mentioned notable ones are also:
If none of these work for your needs, you asked how best to implement this, so here's one way it could be achieved with some customisation...
Your subscription products could have a tag to identify them as products that should initiate a subscription when ordered and a term variant option so customers can order the 1/2/3 month variants.
Then on the order paid webhook you could detect the subscription product in the orders line items, add the subscription tag to the customer and activate their subscription in your backend.
Once the billing date comes around you would use the Draft Orders API to create an order and send out the invoice for that order, this is all possible through the Draft Orders API: https://help.shopify.com/api/reference/orders/draftorder I'd recommend tagging the order as a subscription order to separate it from standard orders.
You could then use the order paid webhook again to detect the subscription order has been paid for and update to the next billing date.
The invoice template is available to edit through the notification settings section of the Shopify store.
Of course this is missing some finer details you would probably need in the implementation including an admin area in the accounts section for customers to manage or cancel their subscription but it should hopefully give you an idea of how it can be achieved. It's quite a lot of work and why the subscription apps exist!