This is a delicate situation. I'm afraid there is no real benefit of separating the two flare systems if they are physically so close to each other, as you described. Several things come to mind:
1) Evaluate carefully under which scenarios the relief rates exceed tolerable heat radiation levels, high enough to prevent from safe removal of the second flare tip. These could usually be (but not limited to): blocked outlet for feed stream, loss of power, loss of cooling fluid, or ESD followed with total depressurization of the plant or various sections of the plant (sometimes triggered automatically by the Fire & Gas System). Include any other relief scenario which results in unacceptable heat radiation levels from the other flare, apart from those that I have listed.
2) After thorough identification and evaluation of all these scenarios, discuss with Operations and Safety personnel how they can be mitigated. Perhaps introducing the recycle mode for the entire plant, with inlet and outlet isolation valves closed, can eliminate all process-related upsets. Make sure you cover ALL the scenarios so that no unexpected event can occur in the proposed (temporary) operating mode of the plant.
3) If all this can be achieved without jeopardizing plant and personnel safety, you are left with the ESD scenario (if such exists in plant design philosophy) under which the entire plant or its sections are automatically depressurized via trip signal from the F&GS. Now you have a potentially very hazardous decision to make - to inhibit the F&G system and provide secondary measures (this can involve bringing fire brigade trucks on site, for example). The key thing is to perform a detailed Hazard analysis and see if there is a sound fallback plan for anything that might go wrong. In some companies, this (F&GS inhibition) would not be allowed even for discussion.
Flare maintenance is usually performed when the entire plant is shutdown and depressurized, for obvious reasons. In your particular case perhaps there are some ways around or certain degrees of freedom, but unless you and your co-workers can come up with a 100% proof and safe plan for removal of the flare tip without shutting down and depressurizing the plant, I strongly advise you not to take any risk for events for which there is no fallback plan. Imagine yourself as one of the maintenance workers who will be hooking up the flare tip to the crane slings, with even a smallest possibility to receive heat radiation from the adjacent flare that can kill you or permanently disable you in just a few seconds. This would then be an unacceptable risk, and everyone who participated in organization of this job would lose his credibility in the Oil & Gas industry anywhere in the world.