How do I select the most recent sample UNLESS there is a sample in a certain month?

I have a dataset with sites and dates of sampling. I am trying to keep only the most recent August sample at each site, but if there is no August sample I would want to simply keep the most recent sample (whether it is before or after August). The months range from June-September.

I'm not sure how I can ask r to decide which row to keep based on what is in the other rows.

I've tried option a, which is based on answers to questions on keeping the most recent sample:

att <- att %>%
  group_by(siteID) %>% 
  arrange(desc(SAMPLE_DATE)) %>% 
  slice(1:1)

but that doesn't necessarily keep the most recent August sample.

And option b which is a little messy but I think closer to what I want:

# get duplicates (multiple samples of same lake)
n_occur <- data.frame(table(att$siteID))
dupes <- att[att$siteID %in% n_occur$Var1[n_occur$Freq > 1],]

# keep most recent august sample
m_att <- att %>%
  mutate(keep=case_when(
    siteID %in% dupes$siteID & month(SAMPLE_DATE) != 9 & !is.na(`CHLOROPHYLL A_OW_TOTAL`) ~ "yes",
    !(siteID %in% dupes$siteID) ~ "yes")) %>%
  arrange(siteID,desc(SAMPLE_DATE)) %>%
  filter(keep=="yes")

m_occur <- data.frame(table(m_att$siteID))
m_att <- m_att[!duplicated(m_att$siteID),]

att <- m_att %>%
  distinct()

But this missed some sites and I ended up losing some sites in the filter.

For example, here's a chunk of my data:

SAMPLE_DATE         siteID
1   2020-07-22 LAK20_NY-10001
2         <NA> LAK20_NY-10002
3         <NA> LAK20_NY-10003
4         <NA> LAK20_NY-10004
5         <NA> LAK20_NY-10005
6         <NA> LAK20_NY-10006
7   2020-09-01 LAK20_NY-10007
8         <NA> LAK20_NY-10008
9         <NA> LAK20_NY-10009
10        <NA> LAK20_NY-10010
11  2020-07-07 LAK20_NY-10011
12  2020-07-09 LAK20_NY-10012
13  2020-08-03 LAK20_NY-10013
14  2020-09-01 LAK20_NY-10014
15        <NA> LAK20_NY-10015
16  2020-09-20 LAK20_NY-10016
17  2020-08-09 LAK20_NY-10016
18  2020-07-26 LAK20_NY-10016
19  2020-08-23 LAK20_NY-10016
20  2020-06-14 LAK20_NY-10016
21  2020-09-08 LAK20_NY-10016
22  2020-06-28 LAK20_NY-10016
23  2020-07-12 LAK20_NY-10016
24  2020-09-02 LAK20_NY-10016
25  2020-07-22 LAK20_NY-10017
26  2020-09-30 LAK20_NY-10018
27        <NA> LAK20_NY-10019
28  2020-07-08 LAK20_NY-10020
29  2020-08-04 LAK20_NY-10021
30        <NA> LAK20_NY-10022
31        <NA> LAK20_NY-10023
32  2020-07-07 LAK20_NY-10024
33  2021-09-21 LAK20_NY-10025
34  2021-07-27 LAK20_NY-10025
35        <NA> LAK20_NY-10026
36  2020-06-20 LAK20_NY-10027
37  2020-08-02 LAK20_NY-10027
38  2020-06-07 LAK20_NY-10027
39  2020-08-15 LAK20_NY-10027
40  2020-09-06 LAK20_NY-10027
41  2020-07-18 LAK20_NY-10027
42  2020-08-31 LAK20_NY-10027
43  2020-08-30 LAK20_NY-10027
44  2020-06-21 LAK20_NY-10027
45  2020-09-12 LAK20_NY-10027
46  2020-08-16 LAK20_NY-10027
47  2021-09-13 LAK20_NY-10027
48  2020-09-20 LAK20_NY-10027
49  2020-07-19 LAK20_NY-10027
50  2020-07-05 LAK20_NY-10027
51  2020-08-01 LAK20_NY-10027

Option a would keep the wrong sample for LAK20_NY-10016 (I would want to keep the sample on 2020-08-09, not 2020-09-30). Option b would work for that lake but for LAK20_NY-10025, option b wouldn't include either of the sampling events when I would want to keep the sample on 2021-09-21.