If the order of group ($p$)selected by attacker then discrete logarithm is still hard?
If the attacker can pick, say, an order $p=3$, then it's easy.
However, if the attacker is constrained to pick a large order, that may not be enough. If the attacker is able to select a smooth group order, that is, an order $p$ which is the product of a multiset of small primes, then the discrete log problem is easy (using the Pohlig-Hellman algorithm).
Hence, unless the flexibility that the attacker is granted is more constrained than the question indicates, the answer is "no, the discrete log problem may be easy"