Only specific way to get the url is via the API as its encoded through a scripting placeholder in the email…. API guide here: Payments | Connector API - so to answer the query, no there is no native way in mews to copy and paste the payment link without emailing first.
You would need to essentially create an API call to get the payment link, transmit it to whatsapp to send to the guest. As far as aware there are no huge GDPR considerations for this in terms of asking someone to pay via a link HOWEVER bear in mind that the payment link may give access to the whole reservation as it is linked to the guest portal - I cannot recall if it does, but this is something to consider and check (just send yourself a payment link and see if it gives access to the whole reservation). Also ensure privacy policy etc allows for this use of their phone number. Some integrations may already do this so worth having a look at available integrations that have the whatsapp facility as they may already do this.