SA Sell In-House Broadcasting System to Other Businesses - Legal?

If there is nothing in writing saying the gym owns the IP, then you are on solid ground.

Best however is to say somewhere in your terms of supply that you own all IP and they have a license for its use.

Hmm not 100% sure.

I assume that your method doesn't care if the screen image is free to air or Foxtel and would work even if the Foxtel/free to air signal stopped. If this is the case then whatever the Gym owner does with it (ie show foxtel/free to air etc) as background is not your concern.

Maybe someone with a little more legal knowledge can help here.

Building the hardware, no great problem.

If it's used the way you describe however, then yes, potential problem - for the user.
If you sell the device sold as "use this device to overlay onto Foxtel's content", then yes, potential problem - for you.

I would want to be sure that the gym (or whoever inthe future) is not infringing on Foxtel's copyright
by inserting other content into the feed.

Thing is, if/when Foxtel cuts off their service (or even sues them...) for breach of copyright,
(or for possibly breaching some other T&C about the integrity of the feed),
it's you they will probably complain to, possibly alleging that "you should have told them".
And if you sold the device for exactly that purpose, then you may have a problem.

Not necessarily.

Going only by what you have told us here (missing facts missing, and unstated ifs, buts, and variables not allowed for),
your device itself does not (so far as we can tell) seem any more unlawful than
a common VCR, or a old fashioned radio-cassette device, or an STB that records - all of which are capable
of making infringing copies of things... but are principally intended for lawful use.

Your device could equally be used to insert content into (other) material
in which the gym holds either copyright, or a bona fide licence ("into their own content").
For example, inserting advertising into their own in-house instructional videos? Not likely to be a problem.

Advertising and product literature (manuals etc) often contain cautions
against using devices to infringe copyright.... but it isn't the manufacturer or the seller
who infringes, it's the user.

That said, given that you now (more than) suspect that your device might be used to act unlawfully
(ie infringe copyright), and that it isn't commercially viable to build just one,
it may now be open to you to decide not to manufacture the device at all.