Per the answer given by @creveti-mihai, I looked at the sudoers man page. Unfortunately, as the man page says, it is not possible to preserve LD_* variables. Here is what the man page says:

 Note that the dynamic linker on most operating systems will remove

 variables that can control dynamic linking from the environment of setuid

 executables, including sudo.  Depending on the operating system this may

 include _RLD*, DYLD_*, LD_*, LDR_*, LIBPATH, SHLIB_PATH, and others.

 These type of variables are removed from the environment before sudo even

 begins execution and, as such, it is not possible for sudo to preserve

 them.

So, I'll have to look at specifying the proxy in apt's own configuration files using Acquire::http::proxy and Acquire::https::proxy.