> Source code which is available under a BSD license is available under BSD, full stop. If the software is constrained by a EULA, then it's not actually BSD-licensed.
It says "software", not source code. You can have software that is BSD licensed and not have source available. The BSD license says that redistributions of source code are permitted, but not that source is made available to you.
> Copyright (c) <year>, <copyright holder> All rights reserved.
> Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met:
> Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer.
> Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution.
> All advertising materials mentioning features or use of this software must display the following acknowledgement: This product includes software developed by the <copyright holder>.
> Neither the name of the <copyright holder> nor the names of its contributors may be used to endorse or promote products derived from this software without specific prior written permission.
> THIS SOFTWARE IS PROVIDED BY <COPYRIGHT HOLDER> AS IS AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL <COPYRIGHT HOLDER> BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
Nowhere grants you source code access. It only says that you are allowed to redistribute the source should you gain access to it. So I can write a BSD licensed program, simply give no one source and only binaries, and that's allowed.
Additionally, incorporating BSD licensed code into a proprietary work does not grant you the ability to relicense the BSD code. It's partially BSD licensed, and you as the distributed of the combined BSD proprietary work aren't compelled to give source.
> If you are the copyright holder, you might also issue the software under another license. But there's not much point to dual licensing when one of the licenses is BSD — a downstream user only has to satisfy the minimal requirements of the BSD license to keep their license to use the software, and they need not satisfy more onerous requirements to get a second license to use the same software.
No, you don't get to relicense the combined work. The BSD components remain BSD licensed.