您需要将stats数据框组合到一个包含Date列的数据框中，然后这就是一个简单的连接。
大概是这样的：

# put all the stats data frames in a list
stats = list(May_1 = May_1, May_2 = May_2)
# (in your real case you probably want to use `stats = mget(ls(pattern = ...))`
# where `pattern` is a regex pattern to identify these data frames

# combine them and get rid of the underscores
library(dplyr)
library(stringr)
stats_df = bind_rows(stats, .id = "Date") %>%
  mutate(Date = str_replace(Date, pattern = "_", replacement = " "))

# and join
Game_Logs %>%
  left_join(
    stats_df,
    by = c("Date", "Batter 1 ID" = "Batter ID")
  ) %>%
  left_join(
    stats_df,
    by = c("Date", "Batter 2 ID" = "Batter ID",
    suffix = c(".1", ".2")
  )