Rather than going for AJ raj, why not target an unpopular train? :p
A simpler alternative, in my opinion, would be to divert JP duronto via DEE garib Rath route, give it few more halts and extend to Delhi.
The way it currently runs, it's schedule is bound to 23:00 departure slot in which the three WR duro operate. Shift it to something closer to around Avantika/Lokshakti time, reroute it via ADI-MSH-AII-JP route to arrive Delhi by early evening. This way a relatively less patronized train will, hopefully, get more patronization, the SJ...
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route gets a fast connectivity and duro rake gets better use than present :p

The logic of other members isn't wrong!
See, the point of Rajdhani is to provide fastest connectivity from state capital to national capital, and usually by shortest route. There is a bit of leeway provided in second clause, but the cases don't justify increasing travel time by multiple hours as would be required in your case.
See, a Raj is created for state capital, and as such majority quota is blocked out for end to end passengers; SR bound raj don't follow this as they don't have much E-E passengers but the ones...
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starting from Mumbai won't get that exception. With that rule, hardly any seats would be available for Rajasthan based passengers and thus would hardly provide any use; a Duronto or even better, an SFE would serve your cause better.
The Era of Rajdhani having much shorter travel time is behind us, as many trains are getting 130 kmph sanctions and seeing reduction in travel times now. The route via rajasthan isn't suitable for 130 so not much benefit would be gained by change of state anyways!

Mumbai and Mumbai based rajs are, so as to say, kinda weird. Their user base is rather varied, so as to say, and has lead to success of all 3 rajdhani.
1)12951 and 12953: these raj derive their user base not just from Mumbai but also from Surat, which is arguably amongst the top 10 cities in India. The passenger base from both Mumbai and surat is bound to Delhi, so intermediate booking does not apply. Another unique point from which they derive passengers is Kota, which is a major entrance coaching hub and has many people from and around Mumbai studying there, leaving to significant people traveling that way
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22221: Mumbai suffers from a major problem of having North South connectivity but not east west one. As a result it is inconvenient for people staying on the eastern side of Mumbai (where, owing to the geography of Mumbai a significant chunk actually stays) to travel towards western line. CR bound raj derives a good chunk of passengers from this, along with deriving passengers from Nasik and Jalgaon who earlier had to rely on much slower Mangala SFE or Punjab mail to reach delhi.
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The user base of these trains doesn't have a significant overlap as you might think.