Your tax home is the place where you make your money. So, as far as the IRS is concerned, you home is the apartment. Your house is your second home. There is no tax relief for traveling between your two homes. Most people just move to where they work. Those moving expenses would be deductible.
The fact that you decide to maintain a second home 200 miles away from your tax home is a luxury that you have to pay for yourself.
However, if you have a legitimate business reason for that travel, you could deduct the trips between the two homes on dates when you actually perform business related activities at both places.
Let's say that you have a business in the town where you own the house. You work on Friday, drive 200 miles and work at that business for a time doing things that you can't do elsewhere. Then, on Sunday, you perform business related work before driving back to your job. You may be able to take a deduction for your mileage on those days.
The business would have to be a legitimate, income-producing business. The travel would have to be necessary to the business. You would have to prove that it was necessary to your business for you to travel home each week and that you actually did work at both places on each day that you are claiming a mileage deduction.
Commuting is never considered a deductible business expense, but business travel is.
Now, if you can establish your house (where you live with your wife) as your tax home, and the job at the plant is temporary, you could deduct the travel and living expenses at the plant as business travel expenses. For that, your assignment at that plant would have to have a definite end date that is less than a year from the start date.
You could, under those circumstances, deduct the cost of meals and lodging for those days when you are at the temporary location. You CAN NOT use the per diem allowance for this since you are not paid per diem. You would have to use, and prove, actual expenses. You couldn't write off any rent, utilities, meals or other living expense for the weekends that you spend at home. Somehow, I get the sense that none of this applies to you. The burden of proof would be on you, and the IRS has been looking out for this kind of thing. If it isn't exactly true, don't try it.
You were correct in inferring that the tax code is set up to make sure that you don't deduct your personal expenses. Having those two homes is your choice - even if it seems necessary to you - which makes the cost your responsibility.