To be fair, it IS certainly possible. You may also have to show your card to security as you enter in to T3 coming from T5. They will scan your boarding pass and see you don't have a departure out of T3. That being said, I've accomplished this with an entire family in tow, and it worked out OK. However, you're spending a minimum 1 hour transiting via the shuttle busses between the terminals and security checkpoints both into T3 and back into T5, and making sure you understand where you're going. Signage for this type of navigation through LHR is not intuitive.

Good luck!