I'm not sure on what basis a 'legal battle' might be fought. This appears basic contract law stuff, only slightly complicated by the need for players to be also compliantly registered with the national FA, EUFA and FIFA
Lucumi agreed to work for Bologna Football Club for a period of time specified in the contract negotiated and agreed between them. Bologna FC agrees to pay Person X an amount of money, also specified in the contract, during the aforementioned period. Sunderland AFC requests Bologna to release Person X from the contract so that Lucami can work for them and they offer compensation (the transfer fee) for agreeing to do so. Bologna refuses to agree to the early termination of the contract, which is their right.
Whilst Lucumi can refuse to honour the contract and not work for Bologna (he is not a slave), this would mean he was in breach (or frustration) of contract an liable to be sued for such. However, even if he did so, Bologna could retain his registration and thereby prevent him from working for another football club. 'Unfair restraint of trade', which was the basis for the Bosman ruling, would not apply because (a) it would not be unfair for Bologna to expect the contract to be honoured and (b) Lucumi could still pursue his trade with Bologna and (c) Bologna were not intending to withhold Lucumi's freedom to have another club register him at the end of the contract.
Additionally, from SAFC's perspective, none of this would be resolved before the teams have to be announced for the game against Brentford.
I tend to agree with the Bologna coach. It would seem to make sense to harmonise the end of the transfer window with the start of the football season. However, Bologna, regardless of their theoretical preferences, are operating in the real world, where currently there is an overlap between the two. They play to the rules as they are, not as they wish them to be.