Mechanism design with multi-dimensional types (here: the willingness-to-pay for each object) is a notoriously difficult problem. Even when you abstract from bundling as you do by assuming that the seller only wants to buy one of the goods.
Your problem is studied by John Thanassoulis in "Haggling over substitutes", JET 2004. Unfortunately, there is no "nice" solution. While it is always optimal to post a fixed price in the single-good case, fixed prices are not optimal with two distinct good and randomization is part of the optimal seller strategy:
Using a model of substitutable goods I determine generic conditions on tastes which guarantee that fixed prices are not optimal: the fully optimal tariff includes lotteries. That is, a profit maximising seller would employ a haggling strategy. We show that the fully optimal selling strategy in a class of cases requires a seller to not allow themselves to focus on one good but to remain haggling over more than one good. This throws new light on the selling strategies used in diverse industries. These insights are used to provide a counter-example to the no lotteries result of McAfee and McMillan (J. Econ. Theory 46 (1988) 335).
Gabriel Carroll looks at a simpler problem in "Robustness and Separation in Multidimensional Screening", Econometrica 2017, where the seller does not know the joint distribution of the willingness-to-pay and considers "worst-case" profits.
A principal wishes to screen an agent along several dimensions of private information simultaneously. The agent has quasilinear preferences that are additively separable across the various components. We consider a robust version of the principal's problem, in which she knows the marginal distribution of each component of the agent's type, but does not know the joint distribution. Any mechanism is evaluated by its worst‐case expected profit, over all joint distributions consistent with the known marginals. We show that the optimum for the principal is simply to screen along each component separately. This result does not require any assumptions (such as single crossing) on the structure of preferences within each component. The proof technique involves a generalization of the concept of virtual values to arbitrary screening problems. Sample applications include monopoly pricing and a stylized dynamic taxation model.
If you punch the title into youtube you will find two of his talks about the paper. His introduction may give you an idea on how other researchers have failed to find a cute solution.
