How does Loyverse handle bar tabs. I want to add a clients name on the tab and print the name on the invoice

Yes.
Basically, You can Use Open-Ticket system https://help.loyverse.com/help/open-tickets.

You can predefine Tickets if You want (common costumers -common needs) https://help.loyverse.com/help/how-use-predefined-open-tickets.

And If You use registered costumers and add them to the ticket/bill then it should print their name on the receipt https://help.loyverse.com/help/customer-info-receipt