How do I turn on "Convert higher bit rate songs to 128 bps AAC".

IThe lectures are an audio stream. You need an internet connection to listen to them.
You cannot listen to them on the shuffle.

And 32 is less than 128 so they would not be converted.

I have gone as far as selecting "convert higher bit rate songs to 128 kbps AAC" on the preference section of iTunes, however, I am unable to turn this on after selecting an item to convert.

With the option to convert the files, you don't select anything to convert.
You simply sync the iPod and iTunes will convert and add the items to the iPod.

I can listen to them on iTunes and when I try to turn on "convert", the option to convert to AAC is grayed out in the Advanced Menu.

The Convert higher bitrate songs to 128 kbps option has nothing to do with the option to convert to AAC. Those are two completely separate functions.
convert to AAC is gray because you cannot convert an audio stream to anything.

To use *Convert higher bit rate songs to 128 kbps*, tick the selection on the iPod summary screen.
Then sync the iPod.
iTunes will automatically convert what you have selected to sync.
The lecture you have will not be converted and it will not be put onto the iPod shuffle since it is a stream and requires an internet connection.

Does this clarify my issue?

It was clear before.
Based on *Kind: MPEG audio stream*, it is an audio stream and you need an internet connection to listen to it.
It cannot be added to you shuffle.

My uni did confirm that we're able to transfer and listen to lectures on iPods.

Then you do not have the correct file in iTunes.