You can use any Matrix client with Beeper, you don’t have to use theirs.
Regardless, there’s nothing stopping you from recreating the same stack using the available tools.
What makes their service unique are the bridges. Download their sources, compile them, and then pair them with any server client combo you want.
If you insist on using their stack, you can still use an OSS client. They chose not to make their client open source as it is, by design, for their service only.
They’re trying to run a business aimed at people who don’t care about open source, and want the same closed source experience they get from their other chat apps but with inter connectivity between third party services.
If you want the latter without any closed source code, you can just go and do that. They’ve released all the important parts.