I would not stumble from one extreme to the other – any sub 500 $ DSO isn’t the right choice to replace a 15 k$ Tek.
My recommendation is the Siglent SDS2354X HD, which is a 12 Bit / 350 MHz / 4-channel MSO with deep 100 Mpts/Ch. Sample memory and extended analysis capabilities, switchable true 50-ohm inputs, providing a VSWR <1.5:1 up to >1 GHz. The actual -3 dB bandwidth is more like 460 MHz.
This scope is reasonably responsive and does not pass out for several seconds as soon as it has anything more demanding to do.
Attached is an example, tailored towards your use case.
SDS2504X_HD_PR_Pair_W10ns_RT1.5ns_d4ns
The signal consists of 10 ns wide double-pulses with 1.5 ns rise time and 14.28 ns period (~4.28 ns spacing) between the individual pulses; the burst period is 100 µs.
The main window shows 4 such double-pulses to illustrate the before mentioned 100 µs period.
The Zoom window shows a detailed view on the pulses and the automatic measurements are pretty spot on with regard to pulse width and rise/fall time. The Rising edge Overshoot measurement ROV reads ~3.2% on average and is not entirely stable, as can be seen at a glance in the measurement statistics (high peak to peak variance and standard deviation) as well as the Histogram Icon (which could be enlarged into a separate window for closer inspection.
The instrument provides an always visible 7-digit trigger frequency counter and the time base has 1 ppm accuracy.