Don't know what might be lurking around in the Tesoros, but in terms of less voltage giving less depth, most if not all modern detectors use a voltage regulator to keep a constant static stable voltage to the circuit board. Meaning, as the battery drains from say 12V down to 11 or lower, it won't make a lick of difference as to what the circuit board sees. If they voltage regulator is designed for say an 8 or 9V output, then that's what the board sees no matter how much voltage you feed the regulator.
And, if the voltage is too high, it can cause the regulator to either burn out or shut down on a thermal overload until it cools off (if it has one). Reason why the output voltage is kept constant and unchanging, is that if it was constantly seeing a moving/chaging/draining voltage, the detector would be drifting and becoming unstable, or at the very least would require some kind of constant user adjustment to stabilize things. It's not like a flashlight, where as the battery drains the bulb gets less bright.
So long as the source voltage (battery) is above the minimum required input to the regulator, then most if not all modern detectors are going to get the same amount of depth when the battery is new versus when it's getting near the low battery alarm. I've played with linear regulators in projects, and the required minimum input voltage is usually say 1 or so volts above the static voltage. In other words, if it's putting out 9V, then the minimum input voltage might be around 10V, where when it drops to around that, the low battery alarm kicks in.
I've seen people try to increase the depth of a modern machine by running more voltage to it. If there is a regulator between the battery and the circuit board, all they are doing is risking frying the regulator. They usually have a rather wide input range they can handle, or at least the ones I've played with in the past did, but there are limits. Linear regulators, for one, bleed off excess voltage buy shunting it off as heat to a heat sink on the top of them.
Should the heat generated by too excessive voltage be beyond what the heat shrink can bleed off as it runs, then the ones I've messed with have an internal thermal overload that will shut it down. Once cooled off, if your lucky it'll work again, but the overloads are meant as a safety and not a switch intended to be relied on, as they have a limited life span and might fail in 1 trip or might fail in 20. You just never know, because they were never intended to be used.
Alkalines are usually the type that are needed to be used in detectors. Those cheaper non-alkaline types are probably a bad move unless a manual says they are OK to use. Rechargeables are a far better option to save money these days. Needless to be throwing out batteries and money with it, when a good nimh will give you hundreds of cycles if not more before it needs replaced.
Used to be years ago Nimhs and nicads had very low capacity compared to what you can get them in these days. Higher capacity= more run time, but with the lower capacities of years ago, people to this day believe they don't offer much by way of run times. Truth is a good 2500ma or higher nimh AA cell will often out run a non-rechargeable AA in run times, or at least they have for me when I've timed it in various devices, such as a pocket camcorder. Another reason why people feel/felt nimhs or nicads don't offer long run times, is that they will self discharge while sitting on the shelf. Charge it completely and days or weeks later it's pretty well drained again even if not even used.
There are now cheap LSD (low self discharge) nimhs on the market that will hold roughly I think 80% of their charger near a year later. Make sure though that you do your research, as some do not live up to their claims nearly as well as others. If using regular non-LSD nimhs or nicads, be sure to re-peak the pack before a hunt if it's sat for say a week or perhaps two or so. I've got some non-LSD nimhs that will go dead in just a couple of days, while others are good a month or two later maybe, still holding enough charge to run something a good bit.
I haven't kept up on the latest in some lithium based cells, but I suspect there are now 9V cells with partial lithium chemistry that might work very well, and even provide much higher capacities than a nimh can in that size 9V casing, as lithium provides higher capacity, less weight, and in a smaller package. But if you do go that route, be sure the minimum voltage drain allowed of the cell, if it has one, is above where the low battery alarm kicks in on your machine.
I run a 3 cell lipo in my machine, to replace 8AAs, which is roughly the same capacity as the stock nimh pack but not nearly as heavy, and the 3 cell pack is very tiny compared to 8AAs. I wired a small plug into my alkaline holder to plug it into, yet I can still move that tiny plug out of the way and run 8AAs again in there if I ever wanted to for some insane reason. Lipos charge in a hour without stressing the pack, but again there is a minimum drain issue- Lipos should never be drained past 3V per cell, for a total of 9V in series. Luckily my machine sounds it's low battery alarm at around 10V, so I'm good.
The fully charged voltage is 12.6V, so right around what a fresh 8AA series pack of regular batteries would be. Another perk to lipos, like many lithium based cells, is that they have just about zero discharge sitting on the shelf. Charge it full, and months later it's ready to rock and roll, but it's not healthy to store them at full charge, so ones I don't plan to use for long periods I put at a specific sleep voltage and store in a fire proof box in my refrigerator.
There are also LifePo4 cells out now that look interesting. The voltage is higher though than an AA, so though they are AA in shape, you can't cram 8 of them into a holder, so they often come with 4 dummy cells to use with them to cut the voltage in half. But, that voltage might still be a bit too high for some machines to handle, so use caution. The minimum voltage drain of those cells would also have to be looked into to make sure your low battery alarm will sound before hand, or by way of keeping track of hours used and then checking with a volt meter to judge cut off points where you should re-charge.
I may look further into what the latest and greatest in 9V lithium based cells might be out there, just to throw one in my Pro Pointer, although that thing gives me almost a year's run time anyway so I never felt the need to use a rechargeable in it.
Final thought- Consider, if you want to stick with non-rechargeables, perhaps the lithium based Energizers. Not sure if they sell those in a 9V but I would suspect they do. I have used them, giving to me from somebody, in various things and must say the extra cost is worth the extremely longer run time they give. They are also very light. 8 of them in an AA holder is a good bit less weight than 8 regular non-rechargeables, like most lithiums are. Lipos have a soft outer casing, so in terms of the weight factor it don't get much lighter than those. For that reason they are big in RC electric plane applications. Smaller, lighter, and higher capacity than other battery types, and able to deliver some rather high current demands.