WASHINGTON—An effort by some Democrats to seek an alternative to President Biden as the party’s nominee appeared to flag on Tuesday, with lawmakers signaling no clear path forward after the president made clear he was dead set against stepping aside. House and Senate Democrats held conference meetings for the first time since Biden flopped at the first presidential debate, a debacle for the party that heightened concerns that the 81-year-old incumbent couldn’t beat former President Donald Trump and may not be fit enough for another four years even if he did win. Lawmakers aired their frustrations with their predicament but...