Hi!
At five-ten the boy will easily cope with a full sized [4/4 as it often called] bass. I am only five-six, and people doubted I'd cope with a 4/4 at the start, so I bought a 3/4 [three quarter sized] first bass, but soon graduated to my teachers 5/4 [over-sized] pair of baroque English basses.
If your son does not aim for pro playing that simplifies the issue. No reason to spend the necessary £ten K in time. So a student bass will suffice, but it must be made of tone-wood not plywood [which is called laminated to make it sound less naff]. Also it must be adequately set-up for bridge height [so distance of the strings from the finger-board] so that it is playable and a pleasure to play. As made, most factory basses are in fact horrible to play for not being set-up [the top cut off the bridge to get the correct height of strings, and the top nut profiled for ease at the bass end of the strings].
That is why - if buying a new instrument - be it a student bass or a professional grade instrument, you need it set up by a professional Luther. I have set my own up under guidance, but it not for the faint hearted.
I am not going to try and recommend individual second hand instruments, but if you find one, I would be happy to look it over for you.
I assume that your German style bow is acceptably good. Pernambuco wood for the stick is more or less essential for a decent bow. Other woods do not have the resilience and relative lightness needed for a good tone to be transmitted from the bowing arm to the strings, and get a good attack, AND sustain.
If you know Thwaites, then do not hesitate to take their advice. They may source you a nice second-hand instrument for a good price. Also consider Turners of Nottingham, but that probably is not a way of buying economically.
I'll post a link to something I would recommend. Tone-wood, and would become a nice instrument after a year or two of playing in.
https://www.caswells-strings.co.uk/product/andreas-zeller-solid-back-double-bass-outfit/
No doubt that you could buy without the bow that comes with it for a better price. Please don't buy a plywood bass. They never run-in, and almost never have any guts or projection. I used to know Hamilton Caswell, and he sold me a very nice French style bow, which I only sold in the last twelve months. Of course he has now retired, but no doubt their comment about fitting out and set-up will allow any instrument bought from them to be a pleasure to use and play rather than a struggle and bugbear with a randomly set-up instrument bought off the internet.
May I offer one additional piece of advice. Clearly your son has the music taped with his previous experience, which is a huge advantage, but please [and I mean please] do get him a few lessons on bow grip, left hand technique and posture, or else he will do what comes easily [naturally you might say] and thus will have real problems with injuries from poor posture and collapsed left hand. This is utterly crucial, if he intends to play in the longer term. Once the correct posture and left hand technique is grasped, no doubt he will over-come the technical issues he needs to to play well if he has a good ear for pitch and is prepared to practice bowing and left hand technique.
If you have any further questions, please post them here, or send me a PM.
Best wishes from George