When you used your 70-200, did you frequently need the extreme 200mm end?
If you were often using 200mm maybe you'll benefit from even more reach. Any variant of the 55-300 comes in under the 5" limit, assuming that they aren't going to force you to cycle through the zoom and focus range. The PLM variant has the fastest autofocus. Aperture of the 55-300 could be an issue as you'll need to use higher ISO to compensate.
If you weren't using 200mm that often, there are many f2.8 lenses smaller than 5" long in the 100-135 range. Do you need autofocus?
- DFA 100 2.8 WR macro is a very sharp lens. The focus range is designed for macro, making AF relatively slow, and MF requires turning the ring a lot, but as the cars will never be very close you should be able to do MF okay. Consider this lens if you'll have occasion to benefit from the macro capability other times; don't get it solely for races.
- If you're okay with manual focus, the modern Samyang/Rokinon 135mm is great.
- If budget is a limitation, you can find used manual focus 135mm f2.8 lenses for less than $50. Many of them are fairly sharp but you'll need to check the review area.
The DA* 200/2.8 seemed like an obvious choice, but its 5.3" length risks falling afoul of an overly zealous guard with a ruler. I've never seen any guard actually measure something.
---------- Post added 08-20-18 at 05:18 PM ----------
Originally posted by MossyRocks ...the only other solution seems to be get a micro 4/3 or bridge camera...
The OP mentioned low light. These smaller sensors tend to give grainy results at high ISO or soft results from noise reduction. They are a great option for daylight sports, though.
The higher end m43 stuff can do okay in lower light. Lens choice is limited because some m43 telephoto lenses exceed the 5" limit. Olympus E-M1 mark 2 plus Panasonic Lumix 35-100mm f/2.8 II (70-200 equivalent) meets the rules but costs around $2700. I certainly wouldn't want to spend that much to meet a venue's camera rule that might arbitrarily change at any time.