Drilled out bubbles almost always show in the final product. This is because the bubble has pushed glass to the side of the bubble, making the glass thicker there and thiner under the bubble. You can fill the bubble with frit, but usually, it is still evident.
I ahve no idea at what temperature you fired the bottom blank, but if you went to fire polish temperature, it is likely the edges of the coaster had begun to draw in, making the edges higher than the rest of the piece. Two ways I know of avoiding this. One is to cut the bottom piece larger than required, and then cut it to size when ready to cap with another piece – avoids the thicker edges. Second is to put small pieces of frit at the corners of the bottom piece to keep the upper sheet raised while the central part falls and pushes the air out – the bubble squeeze.
Third, is to design to avoid the bubbles.
Fourth is to fire the design layer upside down
Fifth is to put the design elements on top of the whole piece.
There are a number of notes on avoiding bubbles on the glass tips blog below.