The most common and most likely answer to this crossword clue is the 5 plus 3 plus 7 letter word HANGS THE EXPENSE
I've seen this crossword clue in The Guardian
H
A
N
G
S
T
H
E
E
X
P
E
N
S
E
Buddy explains!
The straight part
We know that "Suspends the outlay/doesn’t mind what it costs" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case suspends the outlay/doesn’t mind what it costs = hangs the expense.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.