What kind of role could Baelish get with Eddard as the king?

I'm coming up with a fanfic where Ned takes the throne instead of Robert, as Robert died on the Trident (Rhaegar poisoned his sword Dark Sister which Bloodraven was never allowed to take to the Night's Watch) so Ned became king instead. (Ned's grandmother was Vaella Targaryen daughter of Daeron thee Drunken)

Ned completely reorganizes the small council viewing it as having a tendency to go corrupt. New High Council (name of it may change)

Hand of the King Jon Arryn

Lord Chancellor (Lord Hoster Tully)

Master of Laws Stannis Baratheon

Master of Coin Ser Kevan Lannister

Lord Treasurer Lord Tyrion Lannister (sidenote in this scenario he's married to Tysha here, and revealed it at Ned's coronation as she just gave birth to their daughter Joanna, so Tywin can't stop it, and although upset by it, he manages to get Jaime back and Tyrion serves Lannister interests in the capitol so he looks at Tyrion as a tool)

Lord Admiral Lord Gavin Greystark (Stark cadet branch formed by one of Cregan's sons)

Master of Trade Lord Wyman Manderly

Master of Ships Lord Torrhen Greywolf (In charge of construction of new ships, both merchant and for the royal navy)

Master of Works Lord Yohn Royce (basically construction and maintenance of the Kingsroad)

Master at Arms Ser Mandon Moore

Grand Maester Luwin (Pycelle exposed for betraying Aerys)

Captain of the Household Guard Ser Rodrick Cassel

Lord Commander of the of the Goldcloaks Mors "Crowfood" Umber

Along with each kingdom getting two regional advisors for the council.

I was thinking of later on Ned possibly coming up with a new position for overseeing the coinage and royal mint, and moving either Tyrion or Kevan to that role, freeing up on of their roles for Baelish to take, would that make sense, or would it not make much sense?

Also, I was thinking that Baelish also is married off to a Frey here, either Hoster forces him to marry a Frey girl before sending him to the Vale, or on the way to the Vale a depressed Petyr ends up married to one, and thanks to the dowry from Walder Frey he manages to find some sort of resource on the land he rules and is able to grow a town of 30k or so people there in addition to all his less than legal avenues of revenue and being able to make a name for himself in Gulltown.