British English phrase "dot and carry one"

I too had supposed it to be mathematical, but Brewer gives

An infant just beginning to toddle; one who limps in walking; a person who has one leg longer than the other.

According to worldwidewords.org, "dot and carry one" (as used in the book) had implications that the heart skipped a beat. This would coincide with Barrie's answer about what the true meaning is. Referencing a Captain Francis Grose in his Dictionary of the Vulgar Tongue from 1785, the site also offers an alternate explanation:

(Grose also mentions hopping-Giles as another slang term of the time for a person with a limp [...])

The notes provided at the end of an online version of the book seems to have come to the same conclusion defining the meaning as,

An irregular "thump, thump."

(An irregular pulse / heartbeat equates to a skipped beat. In other words, the character could have just said he/she felt heart palpitations.)

I think "dot and carry one" is walking up stairs one step at a time. Stepping on the first step then bringing the other foot to join the first, then repeat all the way up.

Dot and carry one is a technique used when adding numbers in a ledger. In effect you place a dot in the first column and then add a unit to the next column (known as carrying one).
Historically book keepers or accountants using pens would have made a ".1" (dot dash) sound when they updated their accounts, this sounds like a person with a limp or old fashioned prosthetic walking.
The metaphors is therefore likens the sound of a person walking with a limp or prosthetic to the "dot dash" sound a pen makes when a bookmaker adds numbers in a ledger.

My mother always used the expression when referring to someone who was somewhat less than sane, i.e., slightly doolally.

"Dot and carry one" is found in the poem "Gunga Din" by Rudyard Kipling. Gunga Din is an Indian "Bisti", a servant who carried water for the troops, Kipling writes, referring to the Bisti carrying a water flask during a battle, "He would dot and carry one 'til the longest day was done, and he didn't seem to know the use of fear." I presume it meant that he scrambled to keep up with the troops. Jackaline Winspear (in the novel "Birds of a Feather," uses the term to describe her assistant, whose leg had been seriously wounded in WW I, coming up a flight of stairs. This would seem to mean that his step was irregular because he had to favor his wounded leg. Allen Peacock, April 8, 2019.

The phrase "dot-and-carry-one" is used by Mary Stewart in Chapter 6 of her novel The Gabriel Hounds, published in 1967.

She wrote,

Then I recklessly dragged the bed away from the wall. It came across the cracked marble with a dot-and-carry-one screech of broken castors.

I take the "dot-and-carry-one" to mean that the bed moved in a series of bumps and shifts.

So I am thinking the pulse of the character in Treasure Island had become irregular - with longer than usual gaps between the beats, the "dot" being the beat and the "carry one" being the gap between beats.