I think we're all just a bit confused on your goals. If you want to use a PICAXE, it will likely be too slow, regardless of how little you use it. You need fast reaction times and the PICAXE just doesn't have it. I was hoping that my last post would allow you to see you can simply use an AND gate in place of the PICAXE. This is a purely hardware solution, but I think it saves time and money in the long run and meets your goals.
If you really want to pursue a microcontroller solution, look at a PIC and BASIC software. Tracecom and I use PicBasicPro (PBP), but it ain't cheap. There is a free demo version of sorts that will work with certain PIC chips that would probably meet your needs - PBP turns BASIC code into assembly before going to the chip, so it can run, potentially, as fast as C and certainly faster than the PICAXE. Alternately, there is Great Cow BASIC (free) and at Oshonsoft that ErnieM recommends.
So, you can use a PIC and you can still write in BASIC. With a fast oscillator/crystal, you can do the timing on the PIC, but it might be easier to reference a dedicated real time clock (RTC) such as the DS1307 or similar. You can program in software debounce routines or simply use the BUTTON command, so that eliminates the need for a debounce circuit. The PIC would handle the logic, so no need for flip flops or logic gates. You could use the 74C926, but it could get messy depending on how you decide to send the clock signal. You could instead use something like the MAX7219/21 which will display up to eight digits per IC. The advantage here is instead of sending or feeding hundreds of pulses to the 74C926, you simply have the PIC determine what time you want to display once the finish line is crossed then push that value to the MAX7219/21 (won't see the digits rapidly advancing like you would with the 74C926).
Keep in mind this will take time and likely additional parts, so the choice is yours. Based on your experience and comfort level, for this specific application I would strongly suggest a purely hardware solution and avoid a microcontroller.
If you really want to pursue a microcontroller solution, look at a PIC and BASIC software. Tracecom and I use PicBasicPro (PBP), but it ain't cheap. There is a free demo version of sorts that will work with certain PIC chips that would probably meet your needs - PBP turns BASIC code into assembly before going to the chip, so it can run, potentially, as fast as C and certainly faster than the PICAXE. Alternately, there is Great Cow BASIC (free) and at Oshonsoft that ErnieM recommends.
So, you can use a PIC and you can still write in BASIC. With a fast oscillator/crystal, you can do the timing on the PIC, but it might be easier to reference a dedicated real time clock (RTC) such as the DS1307 or similar. You can program in software debounce routines or simply use the BUTTON command, so that eliminates the need for a debounce circuit. The PIC would handle the logic, so no need for flip flops or logic gates. You could use the 74C926, but it could get messy depending on how you decide to send the clock signal. You could instead use something like the MAX7219/21 which will display up to eight digits per IC. The advantage here is instead of sending or feeding hundreds of pulses to the 74C926, you simply have the PIC determine what time you want to display once the finish line is crossed then push that value to the MAX7219/21 (won't see the digits rapidly advancing like you would with the 74C926).
Keep in mind this will take time and likely additional parts, so the choice is yours. Based on your experience and comfort level, for this specific application I would strongly suggest a purely hardware solution and avoid a microcontroller.