I noticed you used a £ instead of a $. So I went to Andertons online and holy crap are cabinets ever expensive in GB! I exceeded your budget and I apologise for that, but I think they have monthly payment plans available for the remaining balance.
There is a secondhand Markbass Littlemark III head for £299, buy that pronto! You aren't ever going to need more than 500 watts, but it is your driver configuration that you have to be mindful of. Were I you, I would buy the 2 8 ohm Fender Rumble 410 mark III cabs they have which are £479 each. If you buy 1 cab your head is only going to push 300 watts into it and I'm not sure that is going to be very much louder than your current Ampeg combo. If you HAVE to stick to your budget, try a TC Electronics K410 on top of a K115 stack, but other than value I can't vouch for what these cabinets sound like.
Finally, 2 pedals are too good to pass on so I would get those as well. Getting the Xotic limited edition BB Andy Timmons preamp pedal for 99 Quid when it normally comes in at £210 is a fantastic option for adding a few more tonal options to an already punchy yet warm sounding head. And then putting a Morley distortion/boost pedal in your effects loop is a great choice for those moments when you want a bump in volume for solos or a little hair on your tone for certain songs (and it wouldn't be too much either as the loop in the LM lll automatically mixes the distorted signal from the pedal together with your clean unaffected signal at a 50/50 ratio). The Morley distortion boost pedal is currently on sale for £69.
Please ensure you try before you buy, all of this gear and particular applications of each piece may sound fantastic on paper, but your active musicman bass may not play nice with the Andy Timmons BB preamp or using 2 cabinets may not be practical for you and you want to completely blow your budget and get a single cabinet such as a 4 ohm Aguilar 4x12 for £1899, so you can roll it in one trip instead of having to make 2 trips carrying a 28 Kg 4x10. I'm done rambling, I hope it was helpful. Best regards!