It's good that you're thinking of this potential complication.
If you construct your conditional prices in a way that is additive (across the conditional attributes) and proportional across the different price points, then you give yourself the best opportunity to be able to fit the data well with just main effects. However, with conditional pricing designs, despite your best efforts to make the conditional prices proportional (across price levels) and additive across the conditional attributes, interaction effects can be significant and needed to model the data well.
And, as you point out, Sawtooth Software's CBC tools are limited to first-order interactions (interactions between two attributes taken at a time). If there are 3-way attribute interactions or 4-way attribute interactions that cannot be accounted for by separate main effects and a series of 2-way interactions between pairs of attributes, then you've got to do something more sophisticated to fit the data well.
There are power tricks for resolving this.
Luckily, your design is just a 4x2x2x5 design, where the last factor (price) has 5 levels. Thus, your first three factors could be collapsed into a single attribute with 4x2x2 = 16 possible levels.
If you respecify your CBC study as two attributes 16x5 (with conditional pricing so that the right prices are shown depending on the 16 possible alternatives), then if you ask the software to give you the interaction between the two attributes, you'd be accounting for the 4-way interaction among all attributes. But, that's a pretty big model! (16-1)+(5-1) main effects + 15x4=60 interaction effects, for a total of 79 parameters in the model. And, many of those interaction effects wouldn't be significant.
Another approach is to again program the survey as a 16x5 design, but be prepared to do some post-hoc data processing in Excel using .CSV files of the CBC data that SSI Web can export. You can play around with collapsing different attributes prior to estimation. For example, imagine you found out that there was a 3-way interaction between the first 2 factors and price, but the 3rd factor could be captured well as a main effect. Then, you'd take your .CSV file that SSI Web exported and you'd reorganize the columns in the spreadsheet to make it look like the respondents had seen one 8 level attribute, one 2 level attribute, and the 5 level price attribute. Then, you'd ask the software to interact the 8 level attribute and the 5 level attribute. That would be (8-1)+(2-1)+(5-1) main effects plus (7x4) interaction effects, for a total of 40 parameters--much more manageable.
Another approach is to treat price as a continuous variable (as a linear or log-linear term) rather than as part-worth (dummy-coded). If the effect of price is fairly close to linear, you can save yourself a lot of degrees of freedom with your interaction terms this way. To do this, you'd again create a 16x5 design (conditional pricing, to show the right prices for each of the 5 levels of price for each of the 16 alternatives). You'd export your .CSV data and use Excel to manipulate the columns. You would code a price column as the actual prices shown to respondents and treat it as a "user-specified attribute" in CBC/HB estimation. Then, you'd have 16-1 + 1 = 16 main effect parameters and 16x1 = 16 interaction parameters, for a total of 16+16 = 32 parameters to be estimated. And, that model captures the 4-way interaction among all 4 attributes.