The BLR is a very handy lever action rifle! Being a lefty, I really like lever actions!
I have a pre-81 in 358 Win that I have had for over 20 years that is my main guiding/backup/horseback/ATV rifle. The previous owner stripped the factory finish and oiled the stock, which I do like. The metal was in poor shape, but after having it tefloned, was like a new rifle again. And I was lucky enough to find a spare magazine for it back then. I too prefer the straight grip to the pistol grip stock.
I shot my first whitetail buck as a young hunter with my friend's Dad's BLR in 308. I have also owned a Belgian in 243 Win (that I shouldn't have sold), and another newer steel receiver 358 Win that would shoot 1/3" groups with handloads that I sold back to my hunting partner when his grandson expressed a desire for one. I am still looking for a 257 Roberts. Have also been contemplating acquiring a laminated T/D version - perhaps in 6.5 CM.
While they are not known for accuracy, they can be acceptably accurate (and better) with their preferred loads, and yes that short, light barrel can heat up quickly, affecting accuracy. When target shooting, give it time to cool between shots and it will not be an issue. When hunting, a couple of quick follow up shots may be necessary, but to be honest, I have never emptied the magazine in a hunting scenario (even guiding) to date so hasn't been an issue.
The triggers are not great to terrible (and not easily fixed), but as with anything, the more you use it, the more you will get to knw it. Mine has terrible creep and over travel, but after getting to know it, I know exactly where it breaks and can keep my shots consistently within a 1" group at 100 yards with its preferred load (Speer 220 gr Hot Cor FP).
I know there was one writer who used a T/D in 300 WSM to take a mountain goat at 600 yards; so can be said to be accurate enough!
As mentioned above, the metal receiver models are preferred to the alloy receivers by many.
I also own 3 Winchester Model 88's and really like them, as they also lock up like a bolt action (and are easier to customize with new barrels/chamberings), but are not as light and handy as the BLR. (And they too have terrible triggers that are not easily fixed). My semi custom in 250 Savage will produce .209" groups with the HSM 100 gr SGK ammo! (It does have the best trigger I have ever found on a Model 88, and the action has been stoned to make it very smooth, and then Cerkoted, which not only protects the metal but adds a bit of its own lubricity)
Best of luck with your new 243! Hope you have many great adventures with it!