Can you do a CU triaxial test on a reconstituted sample and determine your Abar parameter to see if it is contractive or dilative?

Clays that are dilative (over consolidated) induce negative pore pressure when loaded, that dissipates with time. Therefore, their strength decreases with time. Undrained strength parameters represent the strength at the beginning of loading, and are typically less than the long term drained strength . Contrastive clays ( normally consolidated) induce positive pore pressure when loading that dissipates with time. Undrained strength params represent the strength at beginning of loading, but are less than the long term, drained strength.

You mentioned strength parameters for the embankment. Won’t the embankment be recompacted borrow from the native source? That would change soil behavior from its insitu condition.

The determination of over consol or normally consol is usually by a series of oedometer consoldtion tests. There is probably a CPT correlation for that too, but…

There's not enough information in your post to really give an answer (would need to see a sketch of the problem and your test information).

On thing to consider is that high confining stresses suppress dilation and can shift the behaviour to contractive, so if the embankment is high enough it's possible that the stress change from constructing the embanmkment causes soils in the foundation and lower in the embankment to have behaviour that transitions from dilative to contractive.

How are you estimating the state parameter for clays using CPT data? My memory eludes me a bit but I thought that the state parameter correlation from Been and Jefferies / Golder (WSP) that's built into CPET-it is only for coarse-grained soils.

The next step is to obtain samples and define the critical state properties using triaxial test. My question was more focused if there is any difference to define the contractive/dilative behaviour of a soil and the transition between them depending only if it's sand or clay.
The state parameter was defined using Plewes correlation, which was complemented with more recent in-situ data including clayey soils.

You haven't really given enough information to answer the question but, to go back to your original question about undrained strengths, it could be that they are talking about using residual undrained strengths in LEM to assess static liquefaction? otherwise I'm not sure. You haven't really provided enough information about the problem and the site conditions.