If I'm understanding, 22 items are common to the two groups.
The 23rd item is only to be displayed to group1
The 24th item is only to be displayed to group2
You can indeed use a constructed list (following the instructions in the SSI Web Manual at: https://www.sawtoothsoftware.com/help/issues/ssiweb/online_help/index.html?hid_web_maxdifflistofitems.htm)
This will result in the 23 appropriate items to be shown to group1 and the 23 to be shown to group2. Across both groups, 24 items will have been shown.
When you analyze the results with HB, 24 item scores will be computed for each respondent. Group1 respondents will get a "draw near the mean for Group2" for the item that only Group2 saw, and vice-versa. Thus, you'll want to ignore/drop the score for the item that doesn't pertain to each respondent post hoc HB estimation when summarizing the scores.
The nice thing about this approach is that HB can share information to its benefit across the 22 common items, yet also compute item scores for the 1 unique item to each of the respective groups.