The way I've described the process, all items are shown once in Stage 1. Then, in stage 2, losing items are not shown in stage 2. So, yes, in stage 2 not all items are shown. But, each item will appear at least once, because all items are present in Stage 1.
You can estimate the models in Latent Class as well. Each best-worst task is coded as two tasks. In the first task, the dummy codes are all 1s and 0s. But, in the worst task those codes are all multiplied by -1.
You should use "User-Specified" Coding as described in the Latent Class manual. And, you'll need to build your data file by yourself to manage the proper coding.