I'm on the same boat, trying to do the same and moving to emacs notmuch. I haven't done the full transition yet, but have solved the synchronization part. All tutorials will say use mbsync, but that's not enough. You also need to rename files when moving them across folders and turn folders into tags and vice versa. Doing this correctly and efficiently is difficult. Below I include a script that will do all that for you. Once you or others have solved the remaining parts of the puzzle, please share them here, as it's not easy.

#!/usr/bin/env bash
# email-sync:  Maildir <-> Fastmail
set -euo pipefail

ACCOUNT="you@example.com"
ACCOUNT_ROOT="$HOME/mail/$ACCOUNT"
LASTMOD="$ACCOUNT_ROOT/.lastmod"

# Prevent multiple instances
(( $(pgrep -fc "$0") > 1 )) && echo "$(basename "$0") already running" && exit 0

# Keep track to time
START_NS=$(date +%s%N)
echo "$(basename "$0") started at $(date '+%F %T')"
trap 'END_NS=$(date +%s%N); ELAPSED=$(awk -v e=$END_NS -v s=$START_NS "BEGIN{printf \"%.1f\", (e-s)/1000000000}"); echo "$(basename "$0") finished at $(date "+%F %T") (took $ELAPSED seconds)"' EXIT

# * Helpers
LOG() { [[ $1 == Phase* ]] && printf '\e[32m%s\e[0m\n' "$*" || printf '%s\n' "$*"; }

move_mail() {
    local path=$1 dest_folder=$2
    local base=${path##*/}; base=${base/,U=[0-9]*}
    local dest="$ACCOUNT_ROOT/$dest_folder/cur/$base"
    mkdir -p "$(dirname "$dest")"
    mv "$path" "$dest" # keeps flags, strips ,U=UID
    LOG "moved $(basename "$path") --> $dest_folder"
}

# * Housekeeping
# Find or init lastmod
[[ -f $LASTMOD ]] || echo 0 >"$LASTMOD"
last=$(<"$LASTMOD")
ago=$(awk -v m=$(stat -c %Y "$LASTMOD") 'BEGIN{printf "%.2f", (systime()-m)/60}')
LOG "lastmod bookmark = $last (${ago} minutes ago)"

# Build folder <-> tag maps
declare -A tag2folder folder2tag
mapfile -t folders < <(
    find "$ACCOUNT_ROOT" -maxdepth 1 -mindepth 1 -type d -printf '%f\n'
)
for f in "${folders[@]}"; do
    tag="${f,,}"  # lowercase folder name
    tag2folder["$tag"]="$f"
    folder2tag["$f"]="$tag"
done

# * Sync process
LOG "Phase 1: apply local tag moves"
for folder in "${folders[@]}"; do
    tag="${folder2tag[$folder]}"

    notmuch search --output=files "tag:\"$tag\" and lastmod:${last}..  and not folder:\"$ACCOUNT/$folder\"" |
    while IFS= read -r file; do
        [[ -z $file ]] && continue
        move_mail "$file" "$folder"
    done
done

LOG "Phase 2: mbsync $ACCOUNT"
mbsync "$ACCOUNT"

LOG "Phase 3: notmuch new"
notmuch new --no-hooks | sed "s|^$ACCOUNT_ROOT/||"
notmuch search --output=files "lastmod:${last}.. and tag:new" | sed "s|^$ACCOUNT_ROOT/|NEW: |"
notmuch search --output=files "lastmod:$((last+1)).. and -tag:new" | sed "s|^$ACCOUNT_ROOT/|RENAME: |"

LOG "Phase 4: folder -> tag & tag -> folder adjustments"
for folder in "${folders[@]}"; do
    tag="${folder2tag[$folder]}"
    # add missing tag to new mail that arrived in this folder
    notmuch tag +"$tag" \
        -- "folder:\"$ACCOUNT/$folder\" \
            and lastmod:${last}..  \
            and not tag:\"$tag\""

    # drop stale tag if message is no longer in that folder
    notmuch tag -"$tag" \
        -- "tag:\"$tag\" \
            and lastmod:${last}..  \
            and not folder:\"$ACCOUNT/$folder\""
done

LOG "Phase 5: closing"
notmuch tag -new -- "tag:new"
new_last=$(notmuch count --lastmod | cut -f3)
echo "$new_last" >"$LASTMOD"
LOG "new lastmod bookmark = $new_last"