Of course, you can ask the bank (or more precisely, one of its officers) to sign such a contract. But I strongly suspect they won't.
The bank's standard deposit contract will have been drafted and carefully reviewed by their legal counsel. It would be irresponsible of the bank to agree to be bound to any additional contract, especially one that was drafted by a customer, without similarly having counsel review it. Given the hourly rate that the bank's lawyers likely get paid, the cost of this almost certainly will far exceed the value of having you as a customer (particularly as, by your own admission, you are intending to be a rather unprofitable customer).
So if you insist on them signing any contract of yours as a condition of you opening an account, I expect they will just politely suggest that you look for another bank to do business with.
I think most people would instead handle this situation by just carefully reading the bank's standard deposit contract, and verifying, to the best of their ability, that it really does obligate the bank to pay out the bonus and let you close the account without further fee. Then, if the bank doesn't do so in the end, they might complain to the bank's management, or to the regulatory agency which oversees the bank. In an extreme case, they might consider suing the bank, but the amount of money at stake here is probably not enough to make it worth the time and expense of a lawsuit.
There may also be applicable laws in your jurisdiction that forbid the bank from charging a fee to close an account; but you haven't said what your jurisdiction is, so I can't address that.
I personally wouldn't think it likely that there is any "catch" here. The bank is probably expecting that most people, once they have had the account open for six months, won't bother to close it. And now that they have a relationship with this bank, they may think of it in the future when they want a mortgage, CD, investment advice, etc. So the bank may not mind if a few customers take the bonus and leave after six months, since this will (they hope) be compensated by the larger number of customers who stick around.
All that said, if you really don't trust the bank, then perhaps you are better off not doing business with them at all. Your peace of mind is worth more than $1000.