Increased security weaknesses when truncating SHA1
Given message $A$, you have to find message $B$, such that the first 64 bits (say, MSB) of their hashes collide:
$$
MSB_{64}(H(A)) = MSB_{64}(H(B))
$$

This problem is called Second Preimage Search for the function $MSB_{64}(H)$, or Partial Second Preimage Search for the hash function $H$ alone. 

When $H$ is the full round SHA-1, there is no result, theoretical or practical, that would allow you to do it faster than by trying $2^{64}$ (on average) arbitrary strings for $B$. Clearly, your third restriction does not make the things easier, but still, you need to brute-force $2^{64}$ messages to solve the problem.