IMO you are safer compiling from source than you are using old packages from old repositories.
Best advice I can give you is to file a bug report with blackmagic, they need to update their dependencies.
https://www.blackmagicdesign.com/support/
To compile, go to the relevant ubuntu package to identify the package and upstream source
https://packages.ubuntu.com/xenial/libpng12-0
So you want "libpng_1.2.54.orig.tar.xz"
https://sourceforge.net/projects/libpng/files/libpng12/older-releases/1.2.54/
download, extract, compile, install (starting after download and extracting the tar ball).
cd libpng-1.2.54
./configure --prefix=/usr/local
make
sudo make install
To remove again (if needed)
#run within libpng-1.2.54 directory
sudo make uninstall
I understand it seems like a few extra steps, but it is trivial to download and compile in this way and there is no risk of breaking apt.
ldd ./path/to/<program>
to show the libraries the given binary will try to load. Compile or download the obsolete libraries either adjacent to the binary or manipulate envron LD_LIBRARY_PATH to include them. Beware, any software not distributed by your maintainer's package repository is not going to get security updates from thier security team; you're going to have to manually update the library yourself, four times a day is ideal :)