I tried the tape and sewing pieces together but had the same trouble as you, so I use the the hand method. It seems to work the best.

I hand baste then use a long zig-zag stitch on my machine.
It's a bit more work but there's no lumps.

I've been using this for a long time and just love it. Need to use a damp pressing cloth.
https://www.connectingthreads.com/he...All%20products
B0ught mine at Quilt in a Day or Nancy's Notions.

I use wide batting tape, but I use the overlap and cut method to get clean smooth edges. I actually like a bit of a wave in my edges so there isn’t a straight line anywhere, just in case. This means I use a number of small pieces of batting tape instead of one long one, but it is easy.

I have used the batting tape for 100% or 80/20 just a couple of times, poly batting I sew in a cross hatch method, have also done this a few times. I've measured and labeled all my scrap pieces and have made table runners, table toppers, hot pads and place mats with the smaller pieces, for the larger pieces I look in my MANY patterns for one with that approximate size and make that pattern. I have the scrap pieces in small plastic "boxes", with a scrap piece of paper on what sizes are in that box. I have made progress in using pieces but unfortunately, still have a lot to go. LOl

I have used the tape, but I just butt the two pieces together and zig zag like other people said.

Batting tape is what I use for cotton or 80/20. The other is zig zagged. Really though I don't keep many batting scraps. The bin I have is full and I rarely use them. Most go in a pillowslip for the Humane Society to use for pet beds. They don't wash them so need a lot when the old ones get nasty.

Poly I zigzag so I don't melt the poly. wool I sometimes will hand stitch, sometimes fuse(depends on how densely I plan to quilt) and with cotton or cotton blend I use what I was taught was "French fuse", the woven, slightly stretching lightweight fusible that I buy by the yard and cut into 2" strips. It's a bit cheaper than buying the similar stuff as pre-cut roll for this purpose.

45 x 36? Hmm....consider that most quilter's cotton is 42 wide and you could probably piece a simple QAYG strips of varying length in an afternoon.

I'm just saying Project Linus and Quilts of Valor are always heart-warming charities...