BlueZ 5.43 is available in the Stretch repositories, but Jessie only comes with version 5.23.
I'd try the following to get the more recent version:
Add the stretch sources to your sources.list and at the same time tell APT to (still) default to jessie. You'd do the former by copy/paste the jessie line and in the copy change jessie to stretch. You'd do the latter by creating a file /etc/apt/apt.conf.d/40defaultrelease with the following contents:
APT::Default-Release "jessie";
Then do apt-get update to let APT know of the new packages and versions.
To install bluez from stretch/testing, do the following: apt-get install bluez -t stretch. This will install bluez and its dependencies from stretch.
This way you get a nicely packed Debian/Raspbian package, but at the version you want.
Because of the APT::Default-Release "jessie" you'd still install/upgrade package from jessie by default and would only get stretch package when you explicitly request them (with '-t').
There's also a small variation to that and that is apt-get install bluez/stretch and this will only install the bluez package itself from stretch, but the rest from jessie. This may be a 'cleaner' variant as will only install that one package from stretch. OTOH it may cause an installation problem if it can't satisfy it's dependencies from jessie.
In that case I suggest to do it with aptitude like this: aptitude install bluez/stretch.
The dependency resolver of aptitude is more powerful then apt-get's and will likely find a solution to it. It may come up with 1 (or more) 'bogus' suggestions, but just say 'n' then and let it calculate a new solution till you find one that is appropriate.
