
In reality, there is no cake, it disappeared. I'm guessing Big Mom would start destroying everything if she knew there is no cake, so her children told her the Strawhats took the cake for two reasons: to prevent Big Mom from becoming even more angry and strat destroying the country and killing innocent citizens and to make her attack/destroy the Strawhats. They want her to vent her anger on the enemy. What really happened is that Luffy ate the cake, so the Strawhats can't tell Big Mom they they don't have the cake because: she won't believe them and because if they say the truth she will get even more angry at them for eating her wedding cake, appart from disrespecting her, stealing her Poneglyphs, destroying her plans, taking her prisoner, etc.

big mom have been anticipating for the cake so much that she wont accept that her cake were gone. and she saw the cake devoured in front of her btw. and luffy was in the highlight. plus the picture being broken.
the shock switched off all her sanity.
ofcourse, having the culprit right in front of her and within her territory, no way will she stop there.
and also, take note that Big Mom is a big baby brat. she wont stop until you give her what she wants. so these unreasonable reasons are reasonable enough with the current situation.

Perospero is big mom's son so why on Earth will anyone tell that to big mom? It is to their advantage that big mom is chasing the straw hats and pudding is in charge of the cake so from their side there's no problem at all

Ah I guess @Rinchin 's answer will do then
This questions has been honestly bugging so here it is: why can't they just tell big mom they don't have the cake? I mean she's chasing them for the cake so tell her perospero is lying!! Then they'll be rid of that asshole and they can just move on with big mom wreaking havoc on her own island