i get the white coat thing, too, and had to deal with it for both pregnancies.
it was often high at the OB's office (for dd1), so i self monitored at home and kept a log. when i'd sit down to take it i could feel my chest clench, feel the adrenaline of "what if it's high?" hit. so of course it'd be high (say, 140-150/90-100). but if i'd sit there, and check every few minutes a few times in a row, switching arms, so each arm gets that ?5 min break that is supposed to happen before retesting (or else the results are not valid, not that that ever stops the nurse from pumping the cuff back up when she misses the upper #), i would feel the anxiety pass, and get bored, and read the paper or watch TV, and then i'd get a real reading, which would be normal.
thankfully my midwives for dd2 were cool with me taking my own readings (although we did liver/kidney function testing, and one 24hr urine to be on the safe side). when i was with an OB for dd1 they had me doing NST's, which stressed me out to no end, but luckily dd1 passed each one. and both times i delivered a week early, so i didn't have to deal with the added pressure of going "post dates." which of course 50% have to, since 40w is the friggin MEDIAN.
i don't know what my BP was DURING labor for dd2, since it was at home and too fast for the midwives. but with dd1, i took it when first realized i was in labor (at midnight), and it was 165/100. kinda overexcited, much?

so i called in to the hospital, and they didn't act like it was any emergency, but said i should come in early rather than later. once i got to the hospital (~2 hours later, since i had to wait for dh to come home from working graveyard shift), the one time they measured it was ?135/85. i think that was when i first got there (and i was no longer giddy, it was starting to hurt), since later they didn't have time to do much: when they finally got me into a room, they just barely got ready to catch her.
i think when it's white coat HTN, often once mama really gets into "labor land," she's too focused and internal to be reacting to the people around her. like normally i hate needles, but with dd1 i didn't even notice the nurse putting in a heplock, until i felt the blood drip down my arm because she screwed up. and normally that would make me freak out or pass out, but at the time i didn't care a bit, i was too busy dealing with a lightening fast strong labor (4cm to delivery in <2hrs).
so in labor it isn't so labile, and really reflects what is going on inside, instead of reflecting your fear of what someone will make you do if it's high (like that anxiety that authority figures can bring on: being sent to the principle's office, or your boss showing up at your old job).
so it's possible that if you measure at home you will have a better picture of what your body is doing when you're NOT scared. and that info will help you make decisions when pressed by your OB, who from the sound of things is already giving you the heads up that he'll WANT to induce you.
but you also say "this issue has kept me with an OB for care this time," and if that's because you think that a midwife won't take you, well, you might be wrong about that. i thought so too, until i talked to them. all 3 of the midwives i interviewed were willing to work with me on the issue. so if you haven't interviewed any, it's worth a shot.