You can find the repayment amount by the method previously shown here, with the first repayment period extended by the fraction x
representing the extra fourteen days which defers payment from October 1st to October 15th. So x = 14/31
.
Alternatively x
can be taken as a fraction of the average month, i.e. x = 14/(365/12)
. This is effectively how DJohnM's method works, but it runs into difficulty if the payment is deferred to October 31st because the extension is now longer than an average month. Both methods are shown below for comparison.

So first with x = 14/31
. (This would be my preferred method.)
pv = 1000
n = 12
r = 0.1/n
x = 14/31
pv = (c (1 + r)^(-n - x) (-1 + (1 + r)^n))/r
∴ c = (pv r (1 + r)^(n + x))/(-1 + (1 + r)^n)
∴ c = 88.246
Now with x
as a fraction of an average month: x = 14/(365/12)
x = 14/(365/12)
c = (pv r (1 + r)^(n + x))/(-1 + (1 + r)^n)
∴ c = 88.2523
Now by DJohnM's method: calculating an adjusted principal value and using the standard loan formula (shown below).
dailyrate = (1 + 0.1/12)^(12/365) - 1
pv = 1000
adjustedpv = pv (1 + dailyrate)^14 = 1003.662423
adjustedpv = (c - (c + r)^-n))/r
∴ c = (adjustedpv r (1 + r)^n)/(-1 + (1 + r)^n)
∴ c = 88.2523
This matches the previous result, which uses the average month.
Standard Loan Repayment Formula e.g. link, as used in the above calculation.

Checking initial result
With repayments on the 1st of every month, using the standard loan repayment formula.
pv = 1000
n = 12
r = 0.1/n
pv = (c - (c + r)^-n))/r
∴ c = (pv r (1 + r)^n)/(-1 + (1 + r)^n)
∴ c = 87.9159
The OP calculated the repayments as 87.91