I have done a career mode with Everton, I didn't have a problem with Lukaku, kept him on a decent length contract, but I was doing well, winning things and in the champions league. I couldn't get Barkley to sign though, he basically said he was off at the end of the season, he was valued at about £20m, but kept getting bids of £10m and £12m from Chelsea and Arsenal. I ended up doing a swap with him for Ziyech from Ajax, he was valued at £25m, so I did Barkley plus £5m and got him.
If you cant get Lukaku to sign best bet might be to use his value in a swap deal, otherwise you might just get cheeky offers.
One thing I read in a fifa forum is that any money you have at then end of one year doesn't roll over into the next year. So it said to use the excess money to get players you are going to sell onto massive contracts. Then complete the season and start the new one, you can then get rid of the players on the large salaries (you have to swap them, you wont be able to sell them as they never agree terms with the new team!) but you then get their massive wages back, into your transfer fund.