Difficult to keep both happy, Bob. Tulips like a dry(ish) soil after the leaves die off and a good summer baking (you wish) whist your carrots and beetroot will want lots of water at this time. Otherwise I think it would be a nice idea and not to worry about toxins.

the toxins in one plant is maintained in side the cells of the plant. they do not leach across. [Unless we are talking about the chemicals leached by some seeds to prevent germination of others]

Why not take the tulips out and keep them over the summer in a tray of almost dry compost? They would have more chance of survival that way. The tulips in our garden that do best every year are in a bed of poor dry soil.