There is no synonym. Nevertheless you can also translate it “le prix ne peut pas être inférieur à 10€”

“Le prix ne peut pas descendre en dessous de 10€” would be an acceptable translation. “Baisser” could be used as well. “Les prix montent et baissent”.

You mention “excéder” at the end of your question, and I really can’t think of a one-word antonym for either “excéder” in French or “exceed” in English that would adequately contradict/oppose the inherent notion of “to be greater than” contained in those words.

“Dépasser,” on the other hand, doesn’t always imply “greater than.” In certain contexts, especially those like yours containing the notions of “a minimum” and “a maximum” (of something), “dépasser” can actually serve as its own antonym:

“Le prix ne peut pas dépasser ni le maximum de 11€ ni le minimum de
9€”

(I’ve changed your amount [10€] because otherwise your examples essentially mean that the price has to be exactly 10€, i.e., “Il faut que le prix soit 10€).

In the above sentence, “dépasser” means both “no greater than” (11€) AND “no less than” (9€) and it can certainly be argued that the terms “greater than” and “less than” have vastly different, if not opposite meanings.

Unlike with “excéder le minimum,” which to me would always, regardless of context, mean “greater than” the minimum, “dépasser le minimum” can easily be interpreted, in the right context, to mean ”smaller/less than” the minimum.

This “less than” notion of “dépasser” is perhaps best captured in the phrase “un minimum à ne pas dépasser” when used referring to things such human vital signs, car tire treads/air pressure, legal and safety setbacks, mandatory criminal sentences, etc.

Returning to the nuance that I see between “excéder and “dépasser”,” when you think about it, “un minimum à ne pas EXCEDER” is kind of an oxymoron meaning, in essence, “un maximum,” which probably helps to explain why there are absolutely zero hits on Google for it compared to over 20,000 for “un minimum à ne pas DEPASSER” (granted some of which seem to have the “un maximum” meaning, which is certainly attributable to contexts where it’s less “oxymoronic”, but perhaps also to the large number of oxymorons on the Internet).

If “depasser can actually serve as its own antonym” isn’t exactly the kind of answer that you are seeking (and that would be totally understandable), you could perhaps consider some of the following as “kind of” meaning the opposite of “dépasser” (again in contexts involving maximum and minimum limits):

“respecter”: “Le prix doit respecter le maximum de 11€ et le minimum de 9€.” (“Le prix doit respecter les limites maximales et minimales” = “Le prix ne peut pas dépasser les limites établies”)

Or “limiter”: “Le prix doit se limiter entre 9€ et 11€.”

And then, although not a single word, there’s the very formal and fancy:

“en deçà duquel” (below which) = “9€ est le prix en deçà duquel on n’accepte pas.” or the bit less formal

“rester en deçà (de)” (fall short/remain below) =

“Le prix doit rester/être à/égale ou en deçà de 11€.”

“Le prix ne doit/peut pas tomber/être en deçà (de) 9€.”

Finally, if my “kind of”/(depending on the context) “opposites” are likewise not exactly what you are seeking, then I’d have to answer that, no, there is not a one-word antonym for “dépasser” and readily agree with the suggestions offered in the other answers that your interesting question has received thus far.

Interestingly it seems there is no good antonym for exceed in English.