You are correct that when conditional pricing is done, the utilities for the attributes are no longer independent of one another. Thus, the importance scores will also be wrong. For example, if you set a conditional price between brand and price, then the brand utilities each include an intercept related to the average difference in price levels associated with each brand. Market simulations (predictions of choices) are not bothered by this (as the parameters were developed for maximum likelihood fit) and are correct for predicting choices.
The only way to disentangle the results of a conditional pricing design for the purposes of computing independent attribute importances is to re-estimate the model using something outside of Lighthouse Studio, such as our standalone CBC/HB system. And, you'd need to do some extra coding work to prepare the CSV data file for HB MNL estimation. Specifically, you'd need to collapse the Price attribute into a single column for linear price estimation, where that column contained the actual prices shown to respondents (from the conditional pricing table). And, you'd want to rescale those prices in that single X column of the design matrix to be in the singles of units. For example, if your prices shown to respondents were $1000, $2000, $3,500; then you'd want to scale the prices in the X column to be like 1.0, 2.0, 3.5. Otherwise, convergence in HB doesn't work as well. Once utilities are estimated in this way, the brand utilities are independent of the price utilities (the price slope). And, importance scores could then be computed.