What Libyan oil money? Oil cost about a US Dollar a barrel in the mid to late 1930s. Cash strapped Italy probably couldn't afford the infrastructure required to extract it, pipe it to a port, ship the oil, and refine the oil. Saudi Arabia in 1940 only produced 727,000 metric tons of oil or 5,538,000 barrels - which is about .14% of their current production. Don't apply your understanding of current oil pricing and production to the 1930s.
Are we talking about the 1930s or the 1940s-50s, with a neutral Fascist government that never entered the war and thus is still in power? And if we're talking about the 1930s, when exactly, before 1935 or after?
An ideal scenario would be the oil is discovered before 1935. Italy was cash-strapped in OTL, yes - in large part because it was none too neutral and expended three years of its state budget solely for conquering Ethiopia - after 1935.
But if the oil is discovered before 1935, by a regime that is already tendentially neutral, it's entirely possible Mussolini doesn't look for a reboost of his regime and for a bigger empire by invading Ethiopia: he might do so by launching the "battle for the Libyan oil", and look for a
wealthier empire. And invest all that money in infrastructures there.
Then up come the 1940s and the war. You quoted the oil price in the 1930s. Purchasing it where? Do you think the price of oil did not soar, in Europe, with the war? Imagine a neutral Italy that's pumping oil out of Libya, shipping it in neutral tankers to mainland Italy, then selling it to
oil-strapped Germany. I don't think the Germans would pay the same price as the one you could get in Texas.