You'll need a clock with 'on' state that is shorter by 4 game ticks(gt) than its 'off' state.
First, the lamp activates immediately but takes 4gt to switch off. It's an interesting quirk used e.g. to double observer pulses (an observer triggers a lamp with the usual 2gt pulse, an observer watching the lamp will trigger twice - once on activation, and once on deactivation).
Next, the simple hopper clock is really quirky - it's a 15gt period clock with 7/15 and 8/15 duty-cycle respective of which hopper you take the output from (and which of the two is which is locational, directional and in general unpredictable). Since 15 is an odd number, and in the game everything is quantized to gameticks, it is impossible to make the lamp work exactly 50% duty cycle on this specific clock. The best one could do is approximate with 7/15 or 8/15 by generating the correct pulse length (5 or 6gt pulses every clock cycle)
But we can use other clocks.
If we don't want to jump through hoops of activating the lamp with a zero-tick or 1gt pulse (generating these tends to be tricky), powering the lamp with a 2gt signal (1 redstone tick) gives a 6gt activation time, so we need a 12gt (6rt) clock with a 2gt duty cycle.
The lower-left is the simplest approach - a 6-observer clock, 2gt on, 10gt off. The clock on the right is a classic repeater clock equivalent (with an observer-based on-off switch, but you can activate it with a very short flick of a lever too). The upper left is a slower clock - 16gt, with 4gt duty cycle. The extra 4gt of the lamp make it light up for 8gt, out of the clock's 16gt period.
So, generalizing this to any clock - your clock's duty cycle must be 4gt shorter than the perfect "50%" to compensate for the lamp's 4gt off delay.
Where it comes to "compact", the "trimmed" versions of the 6-observer clock probably come close to the smallest one can do. They aren't exactly easy to toggle on/off though so the extended version is preferred (just power a rail permanently). It may be possible to trim it even smaller but it will probably necessitate going outside "standard" redstone mechanics.
Now for a generalized solution for any clock of exactly 50% duty cycle regardless of period:
This is a pulse limiter, that delays the rising edge by 4 gameticks, but removes the output power immediately (within the same tick) as input is gone. This way the initial delay compensates for the lamp delay.
For any clock whatsoever - this would not be pretty. For clocks with odd period this is impossible, simple as that. Can't generate n-and-a-half gameticks pulse. For clocks with even periods, it would require dynamic pulse extenders that generate pulse half the length of the clock's (previously measured) period. It's doable for clocks with period of a multiple of 8gt using hoppers and droppers, albeit the delays coming from the somewhat complex circuitry will cause imprecision that will need to be accounted for, but for clocks with arbitrary period this would probably end up as a mountain of very tricky computational circuitry, especially if the period is not divisible by 4 (in which case the circuitry must measure to a gametick, while most of redstone acts in 2gt increments).
Edit: a silent version of the delayed activation circuit to adjust the timing for arbitrary period 50% duty cycle clocks. Unfortunately without instant-tech (which is only possible with noisy pistons), it's not capable of handling clocks with shorter than 14gt period.
Bonus: Circuitry to handle an arbitrary duty cycle clocks with periods matching 8n+4 gt (arbitrary/variable on the fly, up to 320 seconds period, lowest 20gt)
It's based on generating a pulse half the length of previous cycle, by filling droppers using hoppers during the cycle, and emptying using observer clocks (2x hopper speed) the next cycle; two such pulse generators operate alternating measurement (hopper filling dropper) and generation (dropper emptied) roles every cycle. Unfortunately periods other than 8n+4 gameticks don't match a 'whole item' period of the hoppers (plus toggle overhead) so the result won't be tick-accurate with these - the duty cycle of the redstone lamp will be up to 7 gameticks off (which may not matter that much with periods going into minutes).
