I think for Eagle Claw to be successful you'd have to change so much it would be unrecognisable.
Something that could have been changed before the mission set off was to have a unified chain of command. At that point, all the elements involved were under different commands and so got their own chain of commands version of the plan. How much difference that this on it's own could have made is debatable, but it wouldn't have hurt.
You'd need to change the timing or the weather as well, and/or have the pilots and ground crew trained in working under zero visibility conditions.
That said, the planning was helped by the hostages all being in one location and having good on the ground intel from the CIA, diplomatic sources and also from released personnel. The plan seems fairly straight forward, the timing and the distances involved suggest that they were somewhat optomistic in how long they expected Iranian Militias to respond and how, but by and large that could have come off so long as no AAA could be alerted and moved into the area in time.
In the end, a small disaster could have been much bigger.
I think a lot of Beckwith's plans and recommendations would have gone through anyway, possibly faster as he would have been seen as one of the stars of the operation, so JSOC and the others would have come about anyway.
It would also have meant less business for the SAS.
The SAS made a lot of money for HM Treasury directly and indirectly as they were called on by a lot of Americas normal customers to provide SF training, and the subsequent recommendations for kit. Not to mention the goodwill such trips garnered for other British firms such as BAE, Vickers etc.
So Britian could have been in a worse position had it succeeded, maybe Greneda might not have happened and maybe the Panama op would have been a small SF Mission to lift Noriega instead?