There is no evidence that there were any ulterior motives to the torture of Lumumba and his allies Mpolo and Okito. The simple fact of the matter is that he was hated.

First off though, there is something important that needs to be said about talking of Belgians, US and Congo allies as monolithic groups in this affair, particularly with regards to the torture. Now, all of them were responsible for the murder of Lumumba, of that there is no doubt, but whether they mandated the torture is another matter. The American and Belgian officials, while they helped encourage and motivate the murder and to an extent provided logistical support, in general distanced themselves of the concrete details (although the murders were partially done with the help of local Belgians working for Katanga). They left that up to their allies in the various local governments. So it is unlikely that they were involved with the torture specifically, even if they were certainly partly responsible for the execution.

Whether or not the Congolese politicians encouraged the torture is unknown. Mobutu and Kasavubu had only partial control over their jailers and soldiers, mutinies and the disobeying of orders were not at all uncommon at the time. Now, I doubt they were interested in stopping the abuse of the prisoners but it is entirely possible that the torture, both while they were jailed and when they were being transported to Katanga, was an initiative by the guards and merely tolerated by the higher-ups. The only time where we know that the abuse of the three prisoners was ordered by higher-ups was when, on the 17 of january, not long before the execution, Tshombe (the president of Katanga) and his ministers came to see the prisoners tortured.

The Americans thought Lumumba was a dangerous communist who'd bring the cold war to Central-Africa. The Belgians thought he was a dangerous nationalist and communist who endangered their interests in the region. The Congolese politicians like Mobutu and Kasa-vubu opposed him partially for ideological reasons (Mobutu was strongly anti-communist and thought Lumumba was one, while Kasa-Vubu opposed Lumumba's project of a centralised, united Congo) and moreover he was a dangerous rival for power in what was nearly a civil war in the country. The officials in seccesionist Katanga hated him for his nationalist project of a united, centralised Congo, as well as for ordering a military attack on their state and supporting local rebels. None of them had any interest in preventing Lumumba and his allies from being harmed and some like the Katangese officials actively encouraged it. Finally, the common guards would have shared to a degree the hatred of their superiors for Lumumba and his allies. On top of that it isn't exactly unheard of for such guards to simply take advantage of the situation to abuse their power and torture prisoners, even if they might not have any ideological stakes against them.

source: Death in the Congo by Emmanuel Gerard and Bruce Kuklick

As u/toomanysorrows covered things rather well, and one of the sources I'd used, besides the usual suspects, would have been Gerard & Kuklick, I think I'll add on something a little bit more general on the subject.

And by that I'd like to remind you that the Congolese prison system was notoriously corrupt and notoriously brutal all on its own accord, as most of the colonial penal systems were. In that regard, the Belgian Congolese system was not particularly worse nor better than any of its contemporaries - indeed Lumumba, during his brief first imprisonment, for agitation and mail fraud while he was awaiting trial, was an unpleasant and torturous experience, though not to quite the same degree, naturally. Like much of the Congolese state post-independence, there was a direct line of causality and correlation between pre-independence treatment and post-independence treatment in the penal system. Much of the Congolese bureaucracy, including the military and the penal system, took much of its cues for treatment and ethos in a lot of ways from the brutal colonial era, because there simply was no other system to reference in their mind, and the Belgians had so deliberately under-prepared the country for independence.

That said, Lumumba was not tortured for information, but as a political message to agitators, to the counter-governments and secessionist states in the Congo at the time. Remember, in Kasai you have a government breaking away, you have the Elisabethville government in Katanga trying for independence with UMHK bankrolling Moishe Tshombe alongside the mercenaries and a Belgian-staffed Gendarmerie Katangais to serve as a paramilitary arm of the government. And there's the Gizenga counter-government in Stanleyville that was made up largely of Lumumbists that stood in opposition to the Leopoldville government under Kasavubu.

There were no explicit orders from what I can tell. There's no reference to any orders in Larry Devlin's memoire abuot his time in the Congo, nor is there anything that I could see in The Congo Cables about orders to be particularly brutal to Lumumba, despite there being countless references in both of the books about American distaste for the guy, and of course discussions after the fact in a very indifferent tone about what happened.

The Belgians were likewise giving a wink and a nod, without ever giving orders, and indeed when it was time to kill Lumumba, it was the Flemish-born Inspector of the Gendarmerie Katangais that was part of the party that killed and butchered and then returned the next day to burn and bathe in acid Patrice Lumumba's remains.

From what I can tell, I would suggest that there was a willing indifference amongst the Congolese politicians. They knew all about the conditions, they knew about the treatment, and there was a general sense, an acknowledgement, about what that treatment would be, once orders were given to imprison him. Indeed, on more than one occasion Lumumba and his people out in the Stanleyville counter-government tried to petition for his release, or at least the movement of him to a different facility, but that was never carried out, until the transfer on that midnight flight that flew Lumumba into Katanga where he met his end in a roadside ditch on the property of a Belgian expatriate pig farmer, about 3 miles from the airport outside of Elisabethville.

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Besides Gerard and Kuklick, I've also used:

The Congo Cables. Madeleine G. Kalb. 1982.

Congo: The Epic History of a People. David van Reybrouk. 2014.

Chief of Station, Congo: Fighting the Cold War in a Hot Zone. Lawrence Devlin. 2007.