Please know that the “bounce+....” portion of the FROM e-mail address would show due to a combination of two things. The first thing is that the client (or email recipient) has their email settings (inbox settings) set to show these expanded details. Keep in mind that these emails are being sent from the Recurly servers on behalf of the address that you entered in the FROM field which we share a bit more about in our documentation here. Therefore, all emails would send like that.
However the “bounce+...” email address only shows when the email headers are expanded out (which is set in the recipients email inbox settings). Most customers would not see that, unless they’ve specifically chosen to expand those details in their email settings. That is why your clients would see From: <email@example.com> on behalf of “firstname.lastname@example.org” <email@example.com> since we are signing the SPF/DKIM records and sending these from our servers. Therefore, there isn’t a way to remove this since the emails are being sent from our servers.
More info about email headers can be found here https://www.wikihow.com/Read-Email-Headers
Additionally, some recommendations to go along with setting your FROM email address is that you may also want to consider changing the title of your email notifications in Recurly that specifically identifies you to your clients. It may also be worth it to advise your clients to add recurly.com to their safe list in their email spam configuration. This helps with email delivery, as well.