score:10
The issue won't be the UK embassy, but the airline. With separate tickets, check-in companies usually won't consider you an airside transit passenger, the main reason being you're not "insured" if anything goes wrong with the connection, leaving you stuck in the transit area. As such, they'll most likely deny boarding on the flight to the UK.
While the relevant UK government guidance says DATV holders may be allowed to cross the border in case of disruptions, this is internal guidance for UK officers and hardly for airlines, who instead tend to follow the blanket policy that transit on separate tickets isn't transit.
In your place, I'd get a Visitor in Transit visa, which lets you cross the UK border and stay for max 48 hours. With this, you should have no problems boarding your flight.