As JC has mentioned, I've gotten my fingers into this one because it's a great puzzle to solve.

Still haven't found your Alice Coulthard in the 1841. The Lancaster Alice who has been found in Lancaster 1841 age 70 and who died in 1846 age 75 might match a marriage of William Coulthert & Alice Taylor 11 Oct 1791 St. Mary Lancaster. We are in agreement that this must be a different person, since your Alice Coulthard shows up in 1851 age 82 with her son James and family.

Have now found most of the known children of William Coulthard and Alice Fargher who had children in the IOM 1801 to 1816.

Mary bp 11 Jan 1801 Braddan I think will be the one who married Peter BEATY 16 Nov 1837 Cockermouth. This marriage is in the GRO so a certificate might be helpful.

1841 census at St Helen Street, Cockermouth HO107/155 9/22 pg 2 [brother John Coulthard is a few pages later on the same street]
Peter BATTIE 28 Cotton Weaver Y
Mary BATTIE 35 N Isle of Man
William BATTIE 10 Y
Ann BATTIE 4 Y
>Joseph RAMSEY 28 Cotton Weaver Y

1851 census at 41 Terrace Mile End, Liverpool HO107/2176 825 pg 18 #70
Peter BEATTIE, Head Mar 38 Labourer, b Cockermouth
Mary BEATTIE, Wife Mar 48 Labourer Wife, b Isle of Man
Anne BEATTIE, Daur U 13 Labourer Daur, b Cockermouth
2 lodgers
Alice COULTER, Mother of Peter Beattie, Wid 81, b Isle of Man

This latter person Alice Coulter "Mother of Peter Beattie" is interesting because her name is very close to "Alice Coulthard" mother of Mary. Same age and place of birth too. But this Alice is with her son James. It's almost too much of a coincidence but I don't know what to make of it.

1861 census at 5 Mile End, Liverpool RG09/2653 60 pg 34 #192
Peter BEATY, Head M 48 Labourer, b Cockermouth Cumberland
Mary BEATY, Wife M 58, b Longtown Cumberland [odd. Maybe that’s where they lived when she was young?]
William BEATY, GrandSon Un 2, b Lpool

1871 census in Everton Peter Beattie is 58 Lab b Cumberland, Mary is 68 b Cumberland, Grandson William 12 Scholar b Lancs, Son in law Thomas Miller 30 Joiner b Cumberland, Dau Ann Miller 33 b Cumberland, their son George 3.

Peter died in 1872 and I can’t find Mary after that.

Next 2 children were James 3 Mar 1804 Santon and John 5 Apr 1807 Santon and they've been covered previously.

I think the next child will be Jane Coulthard 29 Dec 1809 Marown, d/o "W_W Coulthard" according to IGI. I've found her too.

A Jane Coulthard married a Henry Goulding 18 Apr 1830 Cockermouth.

1841 census at Charter St, Liverpool HO107/557 18/44 pg 26. Her husband likely at sea.
Jane GOUDING 30 Ind N [FMP indexes as “Gonding” but I think it’s “Gouding”, i.e, they forgot the “L”]
Elsie “ 65 Ind Y [perhaps her mother in law?]
Alice “ 12 Y [great choice of name for first daughter]
Eliza “ 10 Y
John “ 3 Y

JC wonders if "Elsie" is more like "Elise" which is close to "Alice" and therefore this might be "Alice Coulthard" recorded incorrectly. I think this is too much of a stretch, especially as the 12yo Alice is very clearly written. But it's possible I suppose.

1851 census at 25 Warren St, Thornton, Poulton Le Fylde, Fylde, Lancs
Henry GOLDING Head Mar 53 Mariner, b Cumberland Whitehaven
Jane GOLDING, Wife M 39 Mariner Wife, b Isle of Man
Alice BAINES, Daur M 20 Stone Mason Wife, b Cumberland Whitehaven
Eliza GOLDING, Daur U 18 Servant, b Lancashire Liverpool
John GOLDING, Son U 12 Scholar, b Lancashire Liverpool
Jane BAINES, Grandaughter 3mo, b Lancashire Fleetwood.

Jane continues through the 1871 census, which has her age matching to her bapt date 1809.

Final child of William & Alice was Thomas Coulthard bp 14 Jul 1816 Marown and he is more elusive. He might be the fellow listed in the census below, but there are other Thomas Coulthard's about the same age. I couldn't find one born IOM. It's possible that Thomas died early.

1841 census at St Helens Street Cockermouth HO107/155 9/38 pg 1
Ann ALLISON 55 House Keeper Y
>Mary SCOTT 35 Y
William WHITE 30 Hatter I
Thomas ROBERTS 25 Hatter I
Thomas COULTHARD 20 Warehouseman Y
Richard CLUCAS 30 Hatter No [that’s a good Manx name!]

The two 1829 Cockermouth marriages of James Coulthard & Margaret Guy and of his presumed brother John Coulthard & Margaret Young have the common witnesses Richard Coulthard & Sarah Young. I found a Richard Coulthard in 1841 a few pages after John Coulthard on the same street, St Helens Street.
Richard COULTHARD 45 Gardener Y
Ellen COULTHARD 43 Y
Richard COULTHARD 20 Cotton Weaver Y
James COULTHARD 13 Cotton Weaver Y
George COULTHARD 17 Cotton Weaver Y

In 1851 he is age 59 Gardener b Cumb Thursby with wife Eleanor 62 b Cockermouth and son Richard U 31 Cotton Hand Loom Weaver b Cockermouth plus a lodger and a grandson.
In 1861 Ellen Coulthard is 72 mother & widow with James 30 Alice 25 James 7 Ellen 4 and Jane 2, at Cockermouth.

Richard could be an elder brother or half-brother to James & John, or could be a cousin or an uncle. It might be worth chasing him.

The entries for the marriages of Jane (1830) and Mary Coulthard (1837) in Cockermouth would be interesting for witness names, and Mary's marriage cert would provide her father's name & occupation (hopefully William, Joiner).

On the Manx Burial Index, there is a William Coulthard age 53 Braddan 30 Jan 1825. For awhile I though he'd be a candidate to be Alice's husband, but now I think this is the William Coulthard who married Elizabeth Callow 1795 Maughold and had many children in Braddan. Elizabeth is in the 1841 census at Ballasalla Malew and was buried 1850 Braddan.

I hope this is helpful.