
You are probably having trouble because you are using DATETIME_FORMAT() which is turning your date into a text string, which makes it difficult to perform additional math on it. If you remove the DATETIME_FORMAT() from your existing formula, you can use it as the {First of Previous Month} in this formula:

I did longer write-up here in Kuovonne's Guide to Airtable, and included a choice of formulas for both calculating the first of the previous month, and the first Tuesday.
Hello @kuovonne thank you for the explanation, the longer write up was greatly appreciated and solved my problem.
This is what I utilized in the end.
Date Field labelled {Date}
First of Previous Month (Field Name)
DATEADD(
DATEADD(
{Date},
-1 * (DAY({Date}) - 1),
"days"
),
-1,
"month"
)
This output the 1st of the month prior
Then I added another field to find the 1st Tuesday.
First Tuesday of Previous Month (Field Name)
SWITCH( WEEKDAY({First of Previous Month}, "Sunday"),
0, DATEADD({First of Previous Month}, 2, 'days'),
1, DATEADD({First of Previous Month}, 1, 'days'),
2, DATEADD({First of Previous Month}, 0, 'days'),
3, DATEADD({First of Previous Month}, 6, 'days'),
4, DATEADD({First of Previous Month}, 5, 'days'),
5, DATEADD({First of Previous Month}, 4, 'days'),
6, DATEADD({First of Previous Month}, 3, 'days')
)
This outputted the first Tuesday of the previous month.
Thanks again!