Bonjour KnickKnack,
What you describe is a normal behaviour. A lens flare is an effect generated by a light source directly seen through camera lens. So it's normal to have it disappear as soon as the light source is not in the field of view.
What I did in a project to achieve what you want (because I think too that it's not great when the flare suddenly vanishes), is to put glow on the light source. If the glow disappears too when the light source is not visible, create an object with glow placed on the light.
Also, to enhance the effect, create a spot light. Constrain the spot to the first light, and constrain the target to the camera (optional). The idea is that the spot must face the camera. Then, put a volume light in the spot, and make it's spead angle very large, at least 60 degrees.
In the end, you'll have a surrounding "glow" that will be visible even when the light source is not. By the way, you'll have to lower the flare's intensity.
Hope this helps
Salutations - Cheers
Bernard Lebel