The work thing is truly ridiculous. What if it came down to it where you needed a c-section? OBs say no activity for 6 weeks. Would they still make you work and risk you causing more injury to yourself? Do they not to do the 12 weeks unpaid leave? Majority of states do follow those laws, but I know some jobs are different.
At my OB it is standard to check for dilation starting at 36 weeks. I know dilation usually does mean nothing, but its still nice to see if things are starting. I don't understand why your doctor wouldn't check.
I understand why you would want to be induce, but you have to understand the doctors perspective. Induction can be a very long process (took 36 hours of pessary before my water broke on its own, 18 hours later no baby and no sign of her engaging at 10cm and I needed a c-section). This was when I was 39 1/2 weeks, and I was induced for high blood pressure. If your first baby was born at 38, could be a chance for this baby to be early too. But if you're only wanting an induction because of work, I would talk to your employment about giving you more time. Induction could lead to c-section which could lead to longer time off work by the doctors choice, not yours or your work