Setting aside opinions people may have about SAFe and how Agile it really is, there is no conflict between SAFe as it is defined and TDD or BDD. In fact, the official safe write-up for teams suggests that software teams should be using those practices (https://www.scaledagileframework.com/agile-teams/).
SAFe also still prescribes cross-functional dedicated teams just like Scrum does. Now, that said, many organizations have a long journey to get to that point and have challenging practices like people-sharing between teams for a while as they make that change. My best advice is to start in places where the team members are mostly dedicated. I'd also compare approaches like Spec-by-Example, ATDD and BDD and pick one that's going to fit well in your teams. They're very similar in that they all describe functional tests before building the features, but there are some key differences. For example, in BDD, you take it one test at a time like you do in TDD. This is not a requirement of ATDD. You may find that one acts as a stepping stone to where you'd like to get to. Similarly, if one team that is better situated to start using BDD shows a lot of success, it will be easier to get other teams to make the changes to their structure that enable them to start using BDD as well.