There is indeed a dedicated Cartier in T5.

I assume you're trying to get an exchange rather than a refund?

In that case you have a couple of options. One, as mentioned above, is to contact the personal shopper service at LHR and they will accompany you to T3 and back. My understanding is that the service is offered for high spend customers where the shop required is only in one terminal. They might also want "new" money coming in, not a net zero on an exchange. You'll need to be a little economical with the truth when you speak to them.

The other option is to self-transfer to T3 yourself. You will need a considerable amount of time to do this. On arrival at T5, don't enter the terminal but remain airside and follow the transfer route. Get on the buses to T3 - you don't need a boarding pass. On arrival at T3, if questioned at security, say you're visiting the AA lounge. Clear security, go to Harrods. Then do the reverse - get on the airside transfer bus to T5, clear security again, and proceed to your next flight.

If you land at T3, you still need to follow the transfer route, but straight to security rather than having the bus transfer. The rest is the same.

The important thing here is never to go landside because then you can't get to T3. To be safe, you also need to budget around 2.5 hours for this from T5 (probably 2 hours from T3), because you need to clear T5 security 35 minutes before your flight to meet conformance. If you miss this, you'll be removed from your MAN flight and likely will need to buy a walk-up fare.

This isn't a route for the faint-hearted.

As there's a Cartier at T5, which I assume would stock a bigger range, would Harrods refund you in T5 so you could pop over to the Cartier store and re-purchase?