I have both a Lakland DJ and a Marcus Miller Jazz and personally, they are different in lots of ways.
As you say, the slimmer neck, the smaller body and the pre-amp make the DJ a joy to play and gives you a brighter Jazz tone. I have a passive Natural finish DJ going through an outboard Sadowsky pre-amp.
However, there's no getting away from the fact that the thicker, bigger body on the MM Jazz gives you a much deeper Jazz tone for those punchy lows. Don't forget to look at upgrading parts of your bass as part of your budget. I'm just changing the pre-amp in MM Jazz to Aguilar. Hopefully, that will be £70 well spent.
I guess it's a case of getting out there and trying as many as you can!