Jumping right in, the possible workaround for the process that you wish to implement would be to pass the description of the subscription using the subscription_data.description parameter at the time of creating the checkout session.

We're already doing this. You can see this in my previous replies.

Additionally, to pass the same description to the payment Intent and invoice object, you will need to listen to the payment_intent.created and invoice.created events using webhooks so you can create a logic from your end to update their description:

This is silly to have to do and we're not going to implement this. Stripe already causes far too many back and forth API calls as is. They simply need to fix their checkout sessions for subscriptions to either allow for payment_intent_data (this is currently not allowed in subscription mode) or pass along the subscription_data description to the invoices (which they are not doing currently for whatever reason).

Finally, for the one-time payment, you may follow the same process that you're implementing, which is by passing it using the payment_intent_data.description parameter.

We're already doing this.

I don't think there's anything more we can do to help here, sorry. Just consider this a limitation of the platform if they don't plan to improve it. Your alternative is when pulling this information into other software you'll need to have it request the subscription object details as well and pull the description from there.