Good for you to recognize that you can use price as a "base price" and use the continuous random shock just operating on that base variable (often then rounded to the nearest unit, 10 units, 100 units of price, etc.) in our ACBC software. This opens up the opportunity to fit a piecewise price function to the data. Not very many people recognize that this is an option in ACBC as different ways to treat a price attribute.
I haven't seen any research on which approach works better: continuous price per this approach you mentioned, or the typical approach of selecting 4 to 7 discrete levels of price and applying that as a categorical variable in the experimental design...and as either categorical (dummy coded) in the utility estimation or linear in the utility estimation.
Interesting to note that many years ago (early 80s, I think) academics like Dick Wittink found a "number of levels" bias for continuous attributes in conjoint analysis. If you used 2 levels for a continuous attribute vs. 4 levels (but the endpoints covered exactly the same range), price importance was significantly higher in the second case. It seemed that respondents seeing more variation in that continuous attribute somehow paid more attention to it. However, other researchers found that this number of levels effect was really strong moving from 2 levels to more levels. But, moving from more levels (such as 5 or 7 levels) to even more levels showed hardly any increased importance for the continuous attribute. So, number of levels effect may mainly have been an artifact of what happens when moving from an unrealistically low number of levels for a continuous attribute to a typical number of levels.
This relates to your discussion because the use of continuous price instead of discrete levels of price will typically lead to many more unique prices shown to each respondent.
WTP is tricky to do well and we have warned about some of the pitfalls in white papers and in a full chapter of our new book "Becoming an Expert in Conjoint Analysis" which is now available on Amazon. A typical concern is that WTP estimates are often too high, so any change in the design of conjoint analysis that lowers WTP is usually considered welcome and perhaps more realistic. So, the notion that price importance could increase a bit by using continuous price in ACBC instead of discrete levels of price in the experimental design would not be an unwelcome thing to most researchers.