There are two separate questions here, one of which you asked in the text and one of which you asked in the title.
- Can I travel from LHR to LGW to catch another flight without going
through UK passport control? (The question you asked in the text)
No, you cannot travel from LHR to LGW without clearing passport control. Unless an immigration officer were to stay with you for literally the whole trip from LHR to LGW, what would stop you from abandoning your journey to LGW and entering the UK?
- Do I need a separate UK visa to transit/can I transit without a UK
visa? (The question you asked in the title)
First off, you cannot use the British-Irish visa scheme you linked to. It requires that you first travel to the country that issued the visa, in this case Ireland; and moreover the program applies to short stay and visitor visas, which an employment visa is not.
However, it appears that you may you be eligible nonetheless to transit the UK without a visa -- see this UK government link here. I am not sure if the separate airports will pose an issue; in any case you will still have to clear immigration.
It is also worth noting that if you booked this itinerary on separate tickets (i.e. if you have two separate PNRs and bought the two trips separately; it's not clear to me if this is the case) you may run into some issues. If you did do this, your first airline may deny you boarding in India without a UK visa because from their perspective, your final destination is the UK as they can't "see" your trip to Ireland.